Just for the record, the longest win in this endgame is 7 moves:
[FEN "8/8/8/8/p7/8/N7/k1K5 w - - 0 1"]
1. Nb4 a3 2. Nc2+ Ka2 3. Nd4 Ka1 4. Kc2 Ka2 5. Ne2 Ka1 6. Nc1 a2 7. Nb3#
The idea in this position (and other similar positions) is to stalemate the king in the corner; that forces a pawn move and when the pawn reaches a2, you'll need to have ...
In the general case K+N vs K+P is of course a draw - or a win for the pawn if it can promote unhindered.
There is however a famous construction were the knight can force a mate against a king stuck in front of its own well-advanced rook pawn:
[fen "8/3N4/8/8/8/p7/k7/2K5 w - - 0 1"]
1.Nc5 Ka1 2.Kc2 Ka2 3.Nd3 Ka1 4.Nc1 a2 5.Nb3#
There are many ...
It is not possible for king+knight to force checkmate but a helpmate is possible.
Basically the pawn promotes to a bishop, the king moves to the opposite coloured square to the bishop with the bishop next to it and the opposing knight delivers checkmate with the king blocking off the two squares in front of the other king.
Well, the main problem in trying to force the Black king into a corner is that you can't control enough squares for that to happen.
Let us look at KvK, where opposition keeps Black's king from advancing. But Black's king is constantly shuffling, and may escape at any time if White loses a tempo.
When the knight comes in, it spends a move on boxing Black's ...