The word I'm looking for is strictly for an individual event where a player is trying to beat their opponent. In a game like ping pong you might call this event a "serve".
I considered "round" but that seems to imply a bigger part of a game, like at the very end.
Hopefully someone can offer a good word for what I'm trying to express.
Edit: Here is an example sentence:
Just a few more _____ and we should start playing for points.
The exact game should not matter, but lets say I'm playing a racing game.. I suppose for that we would say finishes so "just a few more finishes and..."
Hopefully that clears things up.