queen endgame

I recently played an OTB game that ended in this position with white drawing through repetition.

If black pushes the b pawn to block the check, then I think c6 may cause problems for black. But if he keeps shuffling his king back and forth, can white win because of his extra pawn? Should white play c6 after black moves back to the 8th rank?

It should be Black to move out of check in the FEN.

8/kpp4p/p2q2p1/8/1PP5/P3QP1K/6PP/8 b - - 14 33

White should definitely not be taking perpetual check. He has significant winning chances with the extra pawn. In general, I don't think he should be in a hurry to play c6. He needs to adance his kingside pawns first and get a passed pawn on the kingside, maybe occupying some queenside space with a few key moves on the queenside at the same time.

I think you're right that black shouldn't play b6. White can probably play c5 at an inconvenient time, then. If nothing else, black's king will get exposed.

White has winning chances, black has good drawing chances due to perpetual checks in positions like these. I would have to not be drinking right now to give you a good answer. Good puzzle however.