Well now, as you did yourself I first examined some forcing continuations
1.h6,gxh6 2.gxh6,Kf7 gets me nowhere
Black draws as if there is a single f-pawn on the board.
1.f6 is about the same:
1...xf6 2.gxf6,Kf7 3.Kf5,h6 and again there is no way for white to
promote his f-pawn. 4.Ke5,Kf8 5.Ke6,Ke8 etc.
1.g6 is more promissing:
1.....,h6 (1...hxg 2.hxg6 should win easy, i'll outtempo black)
2.f6+,gxf6 3.Kf5 and I win the f6-pawn back with a passed pawn.
Wait a sec, I know this position: it's a theoretical DRAW.
(As a matter of fact I discussed this position a few weeks ago in the
Hebden pawn endgame)
So there seems to be no winning pawn-break.
What if I move my king. Hm
1.Kd5 looks nice:
After 1...Kd7 I can break through: 2.g6!,hxg 3.f6! winning
After 1...Kf7 2.g6,hxg6 3.fxg6+,Kf6 4.Kd6 black seems in trouble
4...Kg5 5.Ke7,Kxh5 6.Kf7! and I win
(6.....Kg4 7.Kxg7,h5 8.Kf7,h4 9.g7,h3 10.g8Q+,; 6....Kh4 7.kxg7,Kh3 8.Kf7,h5 9.g7,h4 10.g8Q+ 11.Kh2,Qg4)
The pawn on h6 prevents black to defend his g-pawn.
Without it black would even win after Kh6!
In fact this is another key position, that you should be able to recognize:
White e4/kd5 Black e5,Kf4 Whoever has the move loses
But what about 1....,h6!
Now I am forced to play 2.g6,Kf6
And after 3.Ke4 I dont see a win.
But what if I can get to the g-pawn like in the Kf7-variation?
3.Kd6,Kxf5 4.Ke7,Kg5 5.Kf7 yes I am in time to promote.
Finally I can get this last variation by force:
1.g6,h6 2.Kd5 and black has nothing better than Kf6 since after
Kf8 or Kg8 I can drive the King back to h8 before playing f6! followed by Kf7.
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About the second position:
If black had his king on f6 and the white f-pawn is unprotected,
black draws with h5!
So whites first task is to make sure he protects his f-pawn in time:
1.Kg2 being the first move (otherwise black executes his plan)
After that I would start worrying about the position again

.
As a matter of fact it is helpfull to know that if white has an f- or g-pawn on the 5th rank with his king before it, he always wins.