Can White win this knight endgame? [Title "Black to move"]
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[fen "8/p2N1pk1/1pn1p1p1/2p1P2p/5P1P/2P3P1/PP6/6K1 b - - 0 1"]


I thought that I had advantage as White since I was able to temporarily cutt off Black king. I have underestimated Black's counterplay on the queenside with ...Na5, misplayed it and drew.

The more and more I analyze on my own, the more and more I start to wonder: Did I have any winning advantage at all, or was my position just "prettier" ?
You definitely had an advantage, but it's a bit of a challenge to turn that into victory. If you don't use all of your advantages and his disadvantages, it can quickly turn into a draw.

In this position, your main advantage is your Knight - it can jump very easily between very useful squares d7-f6-e4-d6.
Your King can be activated very easily in the next few moves, while black King is quite passive and has limited movement due to your Knight and Pawns (both yours and his) blocking him.
Moreover, your Knight can't be threatened at d7, so it's your decision when to move it (and give his King some breath).

His ...Na5 is a logical choice from his side, but you have nothing to fear, your aim is to get your King closer to the center (Kf2-Ke3) and also put Pawn on b3 (not if he pulls back with his Knight, of course) to disable any chances of his Knight getting through (preferably he plays Nc4 before you play b3, so you play that move with gaining a tempo).

However, the problem you have is that you can't go too far away with your King. Ideally you would come to the Queen side and take his Pawns, while his King watches helpessly; but you can't do that as he threatens Ne7-Nf5, which leaves your g3 Pawn at his mercy if your King isn't there to defend it. Now, you can try to exchange your g3 Pawn (Kf3 or Nf6 followed by g4), but that leaves you with weak h-Pawn which he can attack easily with his King (Kh6-Kh5) or/and his Knight (Nf5). 

The best thing to try (after b3 and after your King is a bit activated on e3-d3) is to get your Knight back to e4 (Nf6-Ne4) to defend g3-Pawn so you can proceed with your King to the Queen side:

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[FEN "8/p2N1pk1/1pn1p1p1/2p1P2p/5P1P/2P3P1/PP6/6K1 b - - 0 1"]

1...Na5 2. Kf2 Nc4 3. b3 Na5 4. Ke3 Nc6 5. Kd3 Ne7 6. Nf6 Nf5 (6...Nd5 7. Ne8+ Kf8 8. Nd6 Ne7 {Otherwise, you win a Pawn with Nc8} 9. a4 a6 {Black is lost otherwise, he either lets your Knight to c8 or your King inside} 10. a5 bxa5 11. Ne4 Nd5 12. Nxc5 Nc7 13. c4 Ke7 14. Nb7 {And you are a Pawn ahead}) (6...Nc6 7. Kc4 a6 8. Nd7 Ne7 9. Nxb6 Nf5 10. a4 Nxg3 11. Kxc5 f6 12. Nd7 {And now you don't have any trouble promoting on the Queen side on time}) 7. Ne4 Ne7 8. Kc4 a6  9. Nd6 Nc6 10. Nc8 Na5+ 11. Kd3 b5 12. Nb6 Nb7 13. Nd7 a5 14. a3 {Now black has no good moves} Kg8 15. a4 c4+ 16. bxc4 bxa4 17. c5 Nd8 18. Kc2 {You can pick up his a-Pawns and push c-Pawns afterwards} 1-0


But note that I didn't perhaps show the best black defense here. Although, to be honest, with his King that limited, he didn't have many options. If he tries to get his Knight to f5 before he provoked your b3, you can play a4-a5 and his defense on the Queen side is ruined. He can only perhaps try to push his Queen side Pawns and pray that he can somehow block you in time, although it's a very hard fight for him:

[StartFlipped "0"]
[FEN "8/p2N1pk1/1pn1p1p1/2p1P2p/5P1P/2P3P1/PP6/6K1 b - - 0 1"]

1...Na5 2. Kf2 Nc4 3. b3 Na5 4. Ke3 c4 5. b4 Nc6 6. Nf6 Kf8 7. Ne4 b5 8. Nd6 a6 9. Ne4 Ke7 10. Nc5 a5 11. bxa5 Nxa5 12. Ne4 Nc6 13. a3 Kd7 14. Nd6 f6 15. Nxb5 fxe5 16. fxe5 Nxe5 17. Kd4 {And now you win a Pawn and have an easy way to victory} 1-0


All in all, your advantage was a bit more substantial than just "prettier" position, at least in my opinion.