Prigozhin, no matter what he says, is running out of people first of all, not shells. Whether he will leave Bakhmut on May 10th (as he had previously threatened on May 5th) is difficult to say, this is a rather political issue and hardly depends on the supply of shells. But shell hunger is indeed a colossal problem for the Russian army. Not because there are no shells, their stocks average about 60% of what it was before the start of a full-scale invasion, the problem is in logistics.
While Russia is fighting according to the books of the 30s, Ukraine acts rationally and if there is a choice to hit a tank or a fuel truck, it hits the fuel truck, and as a result, the tank, which consumes up to a thousand liters per hour, stops after a few hours and can even shoot from a place, because it runs out of battery. Ukraine is simply putting ammunition and fuel depots out of action faster than Russia can replenish them.
According to the regulations, shells should be stored approximately 60 kilometers from the front line, approximately 500 shells per day are needed for the battery, and due to the Hymars strikes, instead of 60 kilometers, storage sites must be moved 150-200 kilometers away. The same applies to fuel, to supply the battalion, approximately 12 constantly working fuel trucks are needed, and they also now need to make trips not for 60 kilometers, but for 200.
In addition to the problem with delivery to the front line, there is also a problem with storage, there should be oil depots where fuel accumulates and where Hymars does not finish. Large hubs, as in Sevastopol, are no longer functioning, and in total, several dozen tank farms remain within reach for fuel trucks, which, in principle, can be used, of which about a quarter were actively working. Half of those actively working in the last few days have burned down. They can be restored in 3-5 months, but if Ukraine can disable them in a matter of days, Russia simply will not have time to restore them.
There are similar problems with shells, they somehow need to be delivered to the frontline zone, and this can only be done massively by rail, there were three such branches in total - Kupyanskaya, Crimean and Mariupol. Two of the three are now actually not used, because. The Armed Forces of Ukraine regained control over Kupyansk during the Kharkiv counteroffensive last year, at the same time as the large railway junction in the south - Volnovakha is being targeted, including by Ukrainian barreled artillery. It is impossible to transport shells through them, only the branch from the Crimea to Melitopol remained more or less functioning, but even there the throughput has decreased. Thanks to the partisans, the rate of delivery of shells from remote regions has also decreased, trains are often derailed, and they do it quite competently, without opening the contacts, so that there is no signal about a violation of the roadway.
All this leads to the fact that Russia has practically lost its advantage in artillery over Ukraine, and if last year at the peak of activity Russia fired several thousand shells per day at Soledar and Severodonetsk, now in Bakhmut at most a hundred and a half to two. For missiles, the situation is even worse, only X-22 remained in the commodity, but they are very inaccurate, they have a standard deviation of 4 km, and for some other missiles there are already less than 30% left, that is, less than the emergency reserve level. This means that each launch of such missiles must be coordinated at the level, if not Putin, then Shoigu.
And this situation will not improve, on the contrary, soon enough the logistics of supplying fuel and ammunition will become so complicated that there will be no more ammunition on the front line than to ensure an organized retreat, as in the case of the retreat from Kherson.