It is not a question of the President's authority to fire anyone but whether the Attorney General can do so. AG Barr gave Mr. Berman a graceful way out by resigning but Breman chose to act like an ass so AG Barr backed up and asked the President to fire him and he did so. You can't be nice to some people and obviously Mr. Breman is one of those people. No harm, no foul. At least he will not be around to cause problems in any other position.
Trump says he is not involved in the Berman firing.
https://twitter.com/Acyn/status/1274435848658759681
Trump on Berman: That’s all up to the AG. AG Barr is working on that. That’s his department, not my department but we have a very capable AG so that’s really up to him. I’m not involved.
If I’m understanding the chain of events in the last 24 hours correctly, it sounds like President Trump had nothing to do with Berman’s original separation at all. I’m not sure what that implies.