What happens if I put a cooked egg in my cake, and then bake it?

As others mentioned, a weird thing to do, but probably harmless. The French are fond of putting chopped up hard boiled egg yolks in things for richness, without much affecting structure.

Does your recipe call for that? I do that when I make Torta Pasqualina, but otherwise, I wouldn't recommend it.

I'm curious why as well, but I can offer that some people put a peeled hard boiled egg inside a meatloaf before baking. So I guess it would be like that, only without the meat.

What sort of cake and why would you want to do that?