Originally posted by XanthosNZ
Kg7 is the only move. If black attempts to push the h pawn you then:
1. Kg7 h4 2. Kf6 h3 3. Ke6 h2 (...Kb6 4. Kd6 h2 5. c7 h1Q c8Q) 4. c7 h1Q 5. c8Q+ . This is a drawn QvQ endgame. If black attempts to pick off the c pawn first then:
1. ...[text shortened]... each your pawn to allow it to promote if black plays Kb6 only.
What about 1. Kg7 h4 2. Kf6 Kb6? Then Black can eat the c Pawn, but White can't get the Black Pawn.
Oh I see this is the same thing as your second line.