What would a generic term be for a puck or ball in a sport?
How about the "mark" - meaning a target?
Use "projectile." This covers anything that is thrown, hit, shot, hurled, tossed, etc.
There isn't one. Just use "ball" and have some comments.
Since the item you are talking about is usually what two or more people fight for mastery over, what about something like object of contention?