White to move. Can he win?

No, ofcourse not. Even if white were able to capture all 4 pawns, there is no mate with a knight and a king. So perhaps white could draw, but there's certainly no win.

Yes, after Ng4.   1...h3.  2. Kf1, f3.  3. Kf2, h3. 4.Kf1, f2. 5. Nxf2#

 Well done, hondaguy! The position is a forced mate for white.

ng4-h3-h2-kf1-nf2?

 And NimzoRoy too!

nice problem.

Interesting problem. I solved after a lot of trial and error, still it's fun and good to come back to for visualization training. Thanks.