Your mom may have a point. I know you may not want to hear this — but your boyfriend pushing you and then having an elaborate apology doesn’t necessarily translate into him being changed. One of the consequences is that he does not get to come right back into a relationship with you.
There is no circumstance where his bad mood and shoving should be tolerated: No circumstance. Right now I would invest your time doing more activities with your friends — and not trying to help him so much. It sounds like he has a lot on his hands, and a lot to deal with. Let him deal with it. If your feelings for each other are genuine he will find a way to honor you — and not treat you so poorly that he pushes you. He has a lot of his own work to do first.