In a poker game Can a player ask for a cut after the deal begins?

This depends on the house rules. It's common in player dealt games, the purpose of the rule is to prevent the dealer from being able to profit by "stacking" the deck (ordering it by slight of hand) to give his opponent a strong hand that loses to the dealers better hand. if players are allowed to ask for a cut at any time, the carefully arranged sequence of cards is broken and the cheat loses all his money before the traditional beating and ejection into the alley.

In a casino game with a dealer, there's no need for this kind of rule as the dealer is a paid employee not involved in the game itself, however players can ask for the deck to be counted down or replaced if they are concerned about its integrity.