What would a generic term be for a puck or ball in a sport?

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?

How about the "mark" - meaning a target?

Good question. In Finnish we call any puck, ball or similar by the general term "peliväline" but there seems to be no English translation for this. I´m surprised!

Literal translation would be maybe "game device" or "sport apparatus"

I´m not sure either is helpful because they´re not really used.