Usually a knight alone can hold against a passed pawn without the help of the king, but the rook pawn is an exception. The problem is that the knight can’t move to the other side of the pawn when attacked by the enemy king. To make the life simpler here is the rule:
In order to defend against a passed rook’s pawn the knight has to occupy any square in front of the pawn except the queening square. If the knight can’t get to this square then the help of the king is needed.
Here are a few interesting examples:
Vishay Anand against MVL in 2016, London Chess Classic
Q: Although it’s not a pure knight against rook pawn ending White can make it artificially. How would you play with White pieces?
A: In the game Vishy played Bf3 and now Black’s rook can’t use the d- or e- file and has no defence against Bxb7.
Black resigned in view of Nxb7 then a6, and Black can’t pawn being promoted. Meanwhile Nxa5 doesn’t require any explanation.
Kim Pilgaard (2432) against Alejandro Moreno (2509) in 2013 – White to move
In the game, White played Ne2 with the idea of Nc1 and Na2, occupying the square in front of the pawn. But he fails to save the game because of 1…Kd2 preventing Nc1.
Q: Can you save the game for White?
A: Yes, like this:
Preventing Kd2 and preparing Ne2 to c1. Let’s check the options available to Black.
a) 1…Kc3, 2. Kd1 then Ne2-c1 when the knight is supported by the king.
b) 1…Kc2, 2. Nd4+ Kc2 3. Ne2 a3, 4.Nd4+ Kc1, 5.Ke2 to d3 in order to support knight on c2 and the game is a draw
c) 1…a3 2.Ne2 Kc2 transposes into option b.