Studies like these aren't meant to be "understood". They are a result of pure calculation. An engine is more than enough to help you with this.
Help with understanding the solution to this endgame study (Grigoriev 1938; lone knight against pawn
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I am trying to understand the solution to this endgame study, and decided to try sharing it to see if writing it down would help me understand it better. For example, I have tried posting the FEN in an engine, and I still can't say I understand the solution. The engine confirms that black has a difficult to see drawing line after 1.Nc7+ Kc4 2.Ne8!.
Of particular difficulty is the following concept: depending on the black king position, white uses a different route to the pawn. The engine confirms that the sequence above is white's only drawing line. This explanation is difficult for me to make sense of. The position and explanation is from Yuri Averbakh's Chess Endings--Essential Knowledge.
The author explains that from e8, "the focal point," the knight can make its way to the b1 square and its surrounding complex of squares via e8-c7-b5-a3 or e8-f6-e4-d2.
I think just writing it down has helped me. What makes the solution confusing is that white relies on certain knight checks to repeat the position or fork the king and the pawn, and the addition of these extra lines complicates the clarity of the central concept: that of two paths to the b1 square from e8, e8-c7-b5-a3 or e8-f6-e4-d2.
Here is the mainline of the solution given by Averbakh. I haven't included exclams to indicate only-moves, since I would need to check them with the engine, or side variations given with the solution. It might be of interest regardless.