I'm not so sure that Putin is able to have Prigozhin eliminated any more. The man has thousands of troops that are clearly loyal to him rather than Russia, and the regular Russian army does not seem to be able to perform at the same level as Wagner (even if you subtract all of Prigozhin's puffery, it still looks like regular Russian MoD underperforms compared to Wagner).

Supposedly Wagner troops are showing up in Sudan the last few weeks. I think Prigozhin is not intending to go back to Russia, until he can go with a few thousand troops (like Julius Caesar crossing the Rubicon). He probably expects to leave Ukraine before Russia loses, then come back to Russia (with thousands of battle-tested troops) in the ensuing chaos.

Putin may or may not see what he's up to. He might have been hoping to exhaust Wagner's troops in Bakhmut precisely to prevent this from happening. Regardless, I don't think Putin wants to reprimand if he can't enforce it, and right now with Prigozhin surrounded by his own troops, I don't think Putin can enforce it. Even less so once Prigozhin is in Sudan.

Plus, a lot of the regular army (e.g. Surovikin) are rumored to like Prigozhin better than Putin. If forced to choose between them, I don't think at this point that Putin can count on them choosing him.