Your panzer 3 need to be upgraded to a model with decent frontal armour and (
critically) the short 5cm gun by the desert. The 37mm models cannot deal with UK infantry tanks, exept by close range shots from ambush with up-the kilt shots with APCR. Difficult to achieve in the desert.
(UK cruisers can be dealt with at reliable ranges (1000 yds) with the short 50, but the 37mm models (or thin-tin 5cm models) will be stung by the 2 pdr if any hit).
Brandenbergs are very expensive and in reality rarer than hen's teeth. Scenario-designer use, for special forces missions. More points to buy = more points for the AI to spend (x1.5 in the delay, x2.5 in the attack).
Halftrack infantry is also expensive (from the extra APC cost), and if your tanks cannot kill the enemy AT assets for you, easy to crack open with pop-guns like the 2 pdr/37mm. In the desert, you will be in LOS usually, so when you dismount the enemy will spot your grunts. It's not easy to find a covered spot for dismounting there, unless you use
smoke.
Halftrack infantry will be spotted moving by the enemy miles away in the desert, and so are a bullet-magnet for enemy artillery fires, as with tanks. Leg grunts are not spotted till closer up. The mobility of halftrack infantry is not that useful on flat open desert maps IMHO, unless there is cover (ridges, villages) where you can dismount in shelter. APC infantry is more useful on European terrain (woods and villages more common, even on flat steppes). Plus later war grunts have panzerfaust etc.
If all you need is the mobility - consider remounting one company in fast light trucks rather than the expensive tracks. You lose the supporting pintle Mgs, but they are good for nipping forwards to grab a defensible point where they will fight from. The trucks then retire to a covered position to the rear. Just don't bunch up when mounted - keep the loaded trucks spaced 2 hexes apart, in case of incoming arty. Add a platoon of 3 towed PaK to the truck pzgr coy,
do keep these tows nearby (to shift the guns) if you can find a dip to hide them in say. Keep a section of 20mm armoured cars or light tanks in overwatch nearby to deal with scout cars, but not in the middle of the infantry's position (as they will attract arty fires once revealed).
10x10 Cm mortar sections (2 tubes per element) is a lot of heavy mortars for that size of force. Static mortars will need to relocate a lot to avoid counter-battery fire, if you cannot find patches of rough to deploy in (not too near the baseline, so runaway crews have a chance to rally and return). If an APC can carry the section, consider having some nearby to move them. Can be difficult in a desert map - the enemy may be able to see your mortar line, whatever you do. So consider using SP-mortars or SP-Arty (Sig) instead. or go for off map arty that out-ranges the common support gun of the UK (25 pounder has a decent range). 10cm K18 is good. One battery for a force your size (battalion), 2 at the outside. I usually leave my (by now well experienced) K-18 bty
idle in the first half of the battle to fire counter-battery on 25 pdrs and 4.5 inch.
Generally, I keep any long range on-map mortars silent till the AI reveals his mortars and on-map arty, and then thump those positions with concentrations of fire, not spreading single mortars or guns on individual targets, the
whole lot concentrated as a hammer-blow onto a revealed platoon position. (Use a piece of paper to record any other "smoke puff" hex locations seen for later counter-battery attention.). You want to kill or cripple the crew, not just make it run away. (gun positions near the enemy base line
may just need a light peppering, as the running crew will tend to go off-map and so the weapon is out of the game.)
If the enemy abandon an on-map arty piece, then adjust fire behind it to encourage the crews to depart further, and/or run off the map. If you simply switch targets on a runaway, then they are more likely to recover from their headache and return. So fire some rounds where you suspect the retreating crew to be, and further encourage their decision to depart.
Gamey tactic #223 - buy 1 or 2 cheap 81mm mortar teams from support points, and plot them to fire in the pre-game turn, and
keep them pounding the area in the early turns, firing continuously (plot on an approach road perhaps). These should attract AI attention, so giving you a clue as to his arty positions. Don't bother buying any ammo supply for these guys, as they will be getting a severe HE-induced headache

!.
In the desert the MMG is super-useful,
if visibility is good. Either your core ones (pzgr have some in the coy assets AFAIR), or bought as support. Set them up in overwatch and lash his infantry with these, they have a splash effect, so go for clumps of grunts set 1 hex apart. Fire 2 rounds at 1 clump, move over and suppress another, "spread the joy". Idea is to pin them in place with MMG fires while calling in arty to effect the kill. MMG are very good in the advance vs delay - any outpost infantry your advanced leg scouts see, should be in the field of fire of your over-watching MMG. Spray them, and keep them running away. Your firing MMG will tend to attract AI artillery, see counter-mortar tactics as above. MMG also useful to deal with UK motor infantry in 15cwt trucks. Consider the maxim-type MMG (e.g MG08) over the MG34/42 as tripod MG since they often have 6 hexes more range (30 versus 24) , which can be more useful in desert.
The problem with early war German infantry is the lack of any decent personal AT weapons. The ATR does not cut it, even on tiddly armoured cars and light tanks. They only pick up once Barbarossa starts, and the Nazis get introduced to Mr T-34 and his cousin Mr Klim Vorshilov..
However, there are a few units with gebalte ladung - not a very effective AT weapon, but better than nothing, especially in the early war. I'd much rather have an AT weapon than the second LMG, for the majority of the sections in the rifle platoon anyway. (In Poland and France, the heavy section with several 50mm direct-fire 50mm mortars is useful too, less so when the desert comes round).
For the grenadiers - consider upgrading them to the units with gebalt ladung ASAP, those with SMG are not that useful IMHO. unit #795 has a decent supply of them, and the Gewher 41 (An SLR I think?) and an x2 LMG available from 6/41.
So your tanks have to take up the main anti-tank heavy lifting till you get decent AT personal weapons (gebalt ladung is only marginal). As mentioned above, the popgun armed mark 3 is not fit to cut the mustard by the desert. The 75mm short armed mark 4 is a grunt-basher, not an AT asset. Therefore the L50 5cm gun is
compulsory. Its sabot will kill Matilda and valentine at 350-450 metres, though not that reliably. It is best if you can engage them from some sort of short-range ambush, popping out from behind a ridge or smoke say. Don't stand still and trade shots, as the 2pdr is dangerous till you get at least class 5 armoured mk 3s, and past it when you have the class 7 and class 8 models (which also sees the L60 50mm into tank service). Upgrade your Mk 3 whenever you get offered models with better armour or the long 50 - the UK is stuck with 2 pounder for a
long time, and every little cm helps vs 2 pdrs. When UK get the 6 pdr, you have the long 75mm gun coming into service to supplement the long 50mm.
In reality, the Germans relied on PaK for their primary tank-killing, not the tanks.
Your best AT asset in the early desert actually is the 50L60 towed ATG (and the captured Russian 57mm when it arrives - nice gun that, a staple in any USSR core of mine if I'm playing them). These are
light ATG and so can be pushed 1 hex by their crews - a vital survival aid. Used offensively with some thought, the 50L60 class weapons can be devastating. Use fast light trucks to move them to a depression or similar, and unload out of sight. Then get the crew to push them into ambush position - a rough hex is good. Use a platoon per company fielded. Buy some extra from support points. Quantity counts with ATG. It's essential that enemy infantry is not allowed to close and spot your Pak - so use the halftracks Mgs and grunt fires to keep those at bay.
The 5cm L/60
AP round can penetrate 8cm (Matilda) of armour
reliably to 400m. It's sabot round can punch through 8cm to 500m reliably (look it up in WW2-APCalc). Shooting at under 500m allows for 'critical hits' with well placed shots, if you can get to-hit chance to over 80% (90+% better). Valentines are but class 7 frontally, and 6 on the sides. (Even the later Grant is only class 6 on the front hull, so you can get a * or so damage at 1000m with the gun on those at 1000-1200m if the wind is behind the shot). The 50/L60 is a
very good weapon in the early to mid war, and useful against the Western allies shermans and cromwells to the war's end.
Towed ATG in support of mech infantry is very good, if you can set up in an ambush position ahead of the enemy arrival - behind a ridge, in a village etc. The dismounted grunts provide security for the Pak, they deal with tanks. Your 20mm armoured cars or panzer 2 can deal with British scout cars, leaving the 5cm Pak for nasty old matildas.
The 88mm (and don't forget the captured ex-Russian 76mm - almost as effective and a smaller unit, so less easily spotted) heavy ATG can kill Matildas at a decent range. But the crews cannot shift them, and they
will attract arty. They need motor transport to shift them, but the problem with deployment from a vehicle is that if you drop the gun off in LOS, it is spotted. Heavy ATG therefore are a one-shot proposition in a wide-open desert map. Set them up, and screw the range down to 1 in this case and fire them manually when there is a pack of targets to chop down all at once, this can be devastating. Try not to reveal them early by firing at a single tank. Heavy ATG, if moved, need to be set up in a new covered position in order to get another chance at an ambush - you can do that with smoke screening sometimes. Smoke, plonk the gun line behind it,
after ensuring that no enemy (that you know of) has LOS to the deployment area! - or there
will be tears

- and screw range down to 1 then wait for the wind to blow the screen away.
Don't use the SP 88mm on a truck - only 18 or so of those were produced, so are rather gamey to have in your core.
Towed ATG do best in numbers. The more you have, the more devastating a volley you can fire from your pre-arranged "Pak-Front" concentrated anti-tank ambush. The more you have, the more likely to have some in a decent position, if spread out a bit rather than concentrated in a Pak-Front. The more you have, the more will survive incoming enemy arty and MMG fires etc. use the light trucks to shift these about - do not stay still too long or you will get stonked. Mobile light ATG of the 5cm class used
actively and
aggressively (
hunt the tanks with them, do not just sit passively) are a very useful asset when you get to Barbarossa (and there are usually more hiding spots in non-desert terrain).
Once the tank threat is gone, light Pak are useful infantry guns - and more accurate to boot than a shortie 75mm, at least while the HE remains.
The other nice toy at this time is the
jpz 1. The 47mm Czechoslovakian gun is rather hot with its sabot round at close ranges. Take a section per company and use as an ambush predator - it needs to fire at 250-300m or so, and never stay out in the open as it is rather killable. But deployed behind a hill, or in a dip and coming up to fire, it can be good especially if your supporting infantry first hose down the enemy tanks to ensure they are buttoned up. And it is rather cheap.
Also to be considered is the unit #481 - SdKfz 10/5 PaK5 - Available 07/040 to 12/041 - a portee 50/L60!. Nippy, but vulnerable. It may be better than the JPZ1 so long as it does not get stonked by enemy arty?.
Whichever Sp-AT you use - upgrade to a marder with long 75/6mm variant when available (42 or so).
recommendations:
- Upgrade to a 50mm model panzer 3, like
yesterday. (If you have the save from your battles before the desert then go back to it I would suggest)
- Change brandenbergers to regular grunts, and find the ones with personal AT weaponry.
- Buy 1xK-18 off-map bty, or the captured 25 pounder(8.76 bty)
- Change some/all mortars to SP arty of some description
- Buy 1 platoon of motor-drawn light pak per infantry company
- Buy an Sp-at unit (Jpz1 or sdkfz 10/5) per rifle company
- possibly a swap to light trucks for some grenadiers
- Buy a platoon of MMG per coy if you don't have these already
- Buy a few sections of SPAA. If not needed for cloud-punching then useful for grunt-bashing. You
will need experienced FLAK later in the war - so preserve these guys as much as possible.
You don't mention any core arty observer, but you have one of these, bought right at the start of the campaign so he can get experienced and so bring in fire more quickly, I presume?. (also remember to use A0 as an observer to train him up - HQ elements are arty observers 'lite' anyway).
Cheers
Andy