Perrin was sent off in the first half of Saint-Etienne's 1-0 Coupe de France final loss to PSG after a sliding challenge on Mbappe, who was substituted with a suspected ankle injury.
Puel hoped Mbappe's injury is not serious as he said Perrin was simply trying to play the ball. On Saturday, PSG confirmed a "serious right ankle sprain," leaving Mbappe's Coupe de la Ligue and Champions League status in serious doubt.
"It is a problem of commitment and the quality of these players [at PSG]," he said.
"You think you've got the ball and they've got such mastery that they can make the difference at any moment. That's been the case.
"I think we still have a few management problems where you have to commit yourself while remaining in control of your actions in any situation. It's also about learning – finding the right mix.
"Of course, I regret his injury and I hope it's nothing important concerning Kylian.
"There was no intention to hurt him. The intention was to play the ball."
Neymar's 14th-minute goal proved to be the winner for PSG.