I have a burnout but I’m just back from a long (national) holiday, so I’m insure of asking to take days off, nor if it will work, since I’m supposed to be rested.

On the other hand I was sick almost all days so idk if those days off count as holidays.

I had a little more than two weeks off around Christmas last year. I got sick on the third day, and finally got over it two days after returning.

No, I didn’t come back rested.
I felt more burned out than when I left.

Talk to your manager. You are only 1 person with this kind of a problem. Mgmt/hr is people overseing multiple people having various problems, incl burnouts. I think they have more xp and know-how dealing with it and can advice you better than yourself.

Speak up. Ask for help when in trouble. Turn to experienced people.