Basic thematic points to note here

1. Black cannot attack the pawn from the second rank. The moment he enters the second rank, h4 wins
2. can't attack from the third rank either, since g3 is blocked by the pawn.
3. White can't win by merely advancing the pawn, since black will still be within the square after stepping sideways into the black long diagonal.

So, White's critical move is 2. Kd5 Hence, black must prevent that, and play 1. Kc4 , threatening the walk along the fourth rank and then down to h3, and retaining the calculus against the advancing pawn while preventing the WK from coming down the board. Keep the opposition as white tries to break through, and the pawn will fall one way or the other

Draw

1...Kc4 2. Kd6 Kd4 3. Ke6 Ke4 4. Kf6 Kf4 5. Kg6 Kg4=

Or, White can play h4 at some point. For simplicity, I will just analyze 2. h4, since the important idea is that White can't let the Black king get to the 5th rank and so must maintain the opposition along the 6th rank as in the note below.

1...Kc4 2. h4 Kd4 3. Kd6 Ke4 4. Ke6 Kf4 5. h5 Kg5.

White can also play h4 and h5 early, but after 1...Kc4 h4 2. Kd4 h5 Ke5 and the Black king is in the square of the h-pawn.