I think it depends on if he impales the body of Prigozhin on a spike and displays it in Red Square. If he doesn't do that, at least metaphorically, he's weak in the eyes of the other people in power, who has has brought up around him to believe that might is right. If he does give a very convincing "This is what happens when you cross me," well, he might buy himself a couple of years, but, really, we all knew he was willing to kill for his position. It's just a question of if he will kill your grandchildren, too.
The Soviet Union has moved on from the simple torture of Vlad. Traditions shift, during the early Soviet era a Makarov round to the back of the head in a pig shed would have been appropriate but what's really required is a proper show trial. You know nine defendants in the dock confessing to sabotage, counter revolutionary motives and Trotskyism. My bet is that hell be poisoned like all the others.
Just for fun read Arthur Koestler's Darkness at Noon, my username is taken from the novels protagonist, great novel on the psychology of Stalinist torture.
But yeah Putin is definitely weakened its like when you have your first leadership spill in a political party, even if the leader survives his days are generally numbered.