if black makes a peice he would move his king and check and win the peice

But the diagram says "White to move".

Anyway, like you say, even if it was black to move he would have been lost.

white should move his king, next move put the queen in front of the pawn then bring the king ovr

it's a draw if the black king can get to the corner

white wins by finding any move to stop the pawn (kmoves, Qa1, Qg7...) even if the black king was on f1 or h1 or somewhere and white's king is on, like, a8, then white still wins by forcing the black king in front of the pawn with a well-known procedure.

white wins by moving his king then its check then white plays queen to g1 then black monovers his king until he wins pawn