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. Kg7 Kb6 2. Kf6 h4 (...Kxc6 3. Kg5 taking the h pawn next move) 3. Ke5 h3 4. Kd6 h2 5. c7 h1Q c8Q. Again drawn.
The important part of defending this endgame is to be sure to try to cover both your pawn and the h pawn. You do this by playing Kg7 as you remain only two moves behind the h pawn (so if black tries to take two moves to blockade or eat the c pawn you can pick up the h pawn) and can reach your pawn to allow it to promote if black plays Kb6 only.
Originally posted by XanthosNZWhat about 1. Kg7 h4 2. Kf6 Kb6? Then Black can eat the c Pawn, but White can't get the Black Pawn.
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.
Oh I see this is the same thing as your second line.