What is the best move here?

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rterhart

This is the crucial position in a game I played yesterday:

White has just played Nf3-h4 (strange, I know).
 
I played what SF classes as a blunder: Kg5.
 
My reasoning was: I trap White's knight, forcing White to play g2-g3. I want this, because I had been looking at White's g- and h-pawns. They could create a passer on the h-file. I see this as an important threat. After g2-g3, not only is the threat of the passer gone, but White's h-pawn is suddenly very weak.
 
The game continued:
34... Kg5
35. g3 Kf6
Job done, the Black King returns to the centre of the board to deal with White's Rook.
 
But as I said, according to SF, this is entirely the wrong idea.
 
So now I have two questions:
  1. What *is* the correct move for Black in this position?
  2. Am I wrong to worry about the threat of the passer on the h-file?

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Edit: I think I have found the correct move.

tygxc

34...Kg5? is a missed win and leads to a draw.
34...Nb6 35 Rxc5 Na4 36 Rc7 Nc3+ should win for black. The black knight and rook are active. The white Nh4 is passive so white practically plays with a knight less. The white rook stands badly in front of its pawn c4.

rterhart

Apparently, that is the correct way forward, but I'm left with the thought: how am I ever supposed to find that?

Most of the time, when someone  analyses a position of mine, they explain the correct idea and I think: of course! How did I not see that?

But here, even when the correct idea is shown by both tygxc and SF, I still don't believe it. I'm supposed to look 3-5 moves ahead, and somehow understand that my increased piece acitivity is worth more than a. White's extra pawn, b. White's misplaced Rook (and Knight), and c. the treat of a passer on the h-file.

Sorry, but no.

RAU4ever

Well, I feel #2 is influenced a bit by an engine. I can see that black gets active, but obtain a winning advantage? I would expect a few more moves before reaching that assesment.

So how can we find these moves anyway? Well, first off, activity matters a great deal in an endgame. Right now your pieces are not doing anything and they are bound to the c-pawn. It isn't uncommon at all to sac such a pawn to get your pieces active.  Here it looks particularly good as the white king will have to retreat and white has misplaced his knight. 

Let's look at a line like: 37. Kf1 (after Nc3+), Rd1+, 38. Kf2, Rd2+  39. Kf1. What are some ideas? Well,  apart from taking the immediate draw. 1 it might be difficult for your opponent to push the c-pawn if you place your rook behind the pawn with Rc2. This way you maybe can tie up all of his pieces or win back the c-pawn, while keeping the activity. 2. And this looks very interesting to me, you can also play for an attack. Bring that king up the board! 39. ... Ke5!? Sample line: 40. Ng6+?, Ke4, 41. Nf4, Kxe3 threatening mate. 42. g3, and now black could try and get the knight to f3. See how fast you can get the attack? That's because black has three pieces in the attack and the white rook doesn't help out defending. Now white can maybe defend with Rc6 or something like that after Ke5, so probably we need to find a different way to get our king in. But you can see the ideas and the mate threats looming. In fact, it's hard to see white getting the time to even push that passed pawn! 

tygxc

#3

"how am I ever supposed to find that?"
Well a combination of thought, calculation, general principles, experience.
Rooks are powerful pieces. In rook endings an active rook often compensates a pawn.
Rooks belong behind passed pawns. Hence Rc5 and pawn c4 is no good for white: the rook is passive.
In this case knights are present too. Nc3 is active. Nh4 is passive. A knight on the rim is dim.