Can the king catch the pawn?
Python 2, 53 40
lambda x,y,p,q:y-2&lt;q&gt;=abs(x-p)+q/7+y/8*5

The king has coordinates (x, y) and the pawn (p, q).
There are three significant cases:

The pawn is on rank 7 and the king on rank 8. To capture the pawn, the king must be on the same file or an adjacent one.
Result: q = 7  ⋀  y = 8  →  |x - p| ≤ 1

The pawn is on rank 7. To capture the pawn, the king must be within six files.
Result: q = 7  →  |x - p| ≤ 6

The pawn is on a lower rank. To capture the pawn, the king must be able to reach the promotion square at most one move after the pawn.
Result: q &lt; 7  →  |x - p| ≤ q  ⋀  y - 1 ≤ q


My solution is just these conditions golfed down. Hopefully there aren't mistakes this time.