Congratulations for winning this endgame and reaching third place
I agree with you that in this position, white did not put up the best possible resistance when he played 1...Kh5 2.f4 here. This blocks in his bishop (the bishop must stay active!), and it puts the f-pawn onto a weak square. You went on to win this very easily by winning a pawn and forcing the transition into an easily winning pawn endgame.
White indeed has good chances of keeping up his fight for a draw, but this is no easy task, especially at that level.
This is a bishop ending with black-squared bishops. Therefore, each side wants to avoid having their pawns on black squares. White is unhappy with having the h-pawn on h4. But white is happy that black's pawns are still on a7, b6 and c5. If black has time, he will happily and immediately move ...b6-b5 on the next occasion, making sure that his pawns will not be left vulnerable or even blocked and fixed on these black squares.
Very concretely, after 1...Kh5, white has the chance to attack the black pawns on the queen side:
2.Bb8! a6 (if 2...a5, then 3.Bc7 and white will win one of the queenside pawns) 3.Ba7! and already, very concretely, black can no longer avoid losing a pawn on the queenside:
a) 3...b5? 4.b4! the pawn on c5 is pinned! 4...Kxh4 5.bxc5 and white even has the advantage now, threatening 6.c6! (6...Bxa7 7.c7 and the c-pawn queens). The c-pawn will become a passed pawn, and black will have to sacrifice his bishop to stop it from queening.
b) 3...c4 4.bxc4 Kxh4 5.a4!? threatening a4-a5, with another pin 5...a5 Now white is quite happy: He has won a pawn on the queenside, so black will no longer be able to create any passed pawn there. In addition, the black pawns on b6 and a5 are now on black squares, so the black bishop will always need to stay with them to protect them. So the danger on the queenside is stopped, and white has only one threat left to deal with: The passed h-pawn should be controllable with Kf1-g2 if necessary.
6.Bb8
White has much better drawing chances than in the game.
I reached this endgame position in a recent tournament game.My Opponent played h4+ and offered my a draw.I refused the draw and continued with 1...Kh5 and after the bad 2.f4 move I had a winning position because I can win the h-pawn and forced the exchange of the bishops by 2...Bf6 3.Bh2 Kxh4 4.f3 Kh3 5.Bg1 Bd4 6.Bf2 Bxf2 7.Kxf2. Then I prepared the creation of a second passed pawn by 8.b5 and my Opponent resigned an I could finish In this under 1500 Tournament with the 3rd Place.
I wonder if Black can win this position when White plays after 1...Kh5 better moves like 2.Bb8 or 2.Kd3. I think my plan should be to exchange the Bishops and win the isolated h-pawn and push the passed pawn . But If the Bishop goes to Bb8 I think I can not force the exchange of the Bishops. I think that this endgame is difficult to win for black. Has someone an idea how I could Improve this position and if Black can win this position ?