“I can’t see any way black can force avoid white’s mating threat” – is not true. After 1. Kh4, black can draw the game (1… Qf2+ 2. Kxh5 Qxh2+ 3. Kg4 Qg2+ 4. Kh4).
Actually after 1. Kh4, black can draw the game by repetition: 1… Qf2+ 2. Kxh5 Qxh2+ 3. Kg4 Qg2+ So, White has to play 1.Kf5 1nd if 1…Qb1+, then 2.e4 and wins, since his threats are very strong.
Q*f7+ Kh8 Ng6++
Hm. This was my first answer. But a minute later I realized that the king i in check.
I can only see 1. Kh4 as viable for white. Let us look at the other options:
1. Kxh5 Qh3++
1. Kg4 Qg1+ and then:
2. Kxh5 Qg5++, or
2. Kf5 Qg5++, or
2. Kh4 Qg5+
3. Kh3 Nf4++ etc.
1. Kf5 Qb1+
2. Kg4 and degenerates to the case above.
What remains is Kh4. I can’t see any way black can force avoid white’s mating threat.
Slight correction in my previous analysis:
“I can’t see any way black can force avoid white’s mating threat” – is not true. After 1. Kh4, black can draw the game (1… Qf2+ 2. Kxh5 Qxh2+ 3. Kg4 Qg2+ 4. Kh4).
Actually after 1. Kh4, black can draw the game by repetition: 1… Qf2+ 2. Kxh5 Qxh2+ 3. Kg4 Qg2+ So, White has to play 1.Kf5 1nd if 1…Qb1+, then 2.e4 and wins, since his threats are very strong.
1. Kf5
The best white can hope for is a draw. Kf5 loses to Ng7.