Case 1: Both kings are away from the passed pawn: IF Black can get into the square, it is a draw.
Case 2: The Black King is in front of the pawn: depends on the position of the White King
Case 3: Black's king is outside the square, but White has a rook-pawn - If Black's king is in front of the pawn, it is a stalemate, and in some other cases, White's king ...
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.