Mail app keeps opening up with "failed to log-in" message

So for quite some time now, I have had an issue, where the mail app will pop up on the screen a couple of times a day (interrupting whatever I was doing), and in the right corner of the app, under the search bar, it says "failed to log-in". Then just a few seconds after popping up, it will disappear again, as it must have managed to log-in. And thus the problem is solved .... until it pops up again an hour later - and it keeps going like that.

I just got a new MacBook, but it is still happening, so I don't think it should have anything to do with the computer itself.

For info:

I have two gmail accounts connected to my mail app.

Hope that anyone will be able to help me solve this mail-mystery 🧐