Will a small piece of cellphone screen that is missing emit more radiation?

It's non-ionizing radiation and in a tiny amount even if you took the whole antenna out (pretty much by design; otherwise you'd have to carry around a car battery in order to keep the phone running for the day).

Also the display isn't designed to be a RF shield, I don't think the phone would go above the nominal SAR levels reported on the manual even if you took the screen or part of the screen out, but someone more expert than me may correct me if I'm wrong on how the measurement is taken. In any case the difference should be minimal / insignificant.