This game was played against someone at my local chess club Friday night, and I think it was my most accurate endgame play ever. I'm just looking for anywhere I could have improved my play.

33 ... Rxf3 is better.  White's a-pawn won't be a factor.  While he's advancing his a-pawn, you'll pick up his remaining Kingside pawns, get your rook to the a-file, and take the pawn when his King gets close enough to support it.  His Rook won't be any challenge to your four pawns.  With the move you made, White will be able to hold out a lot longer with good play.

After 33 ... Rc6, you played well, but White made more than a few mistakes to help you along.

At the time, and now still, I thought that Rc6 was a cleaner win. The plan of laterally holding my pawn while keeping white's king contained just seems easy to me. I don't doubt you when you say Rxf3 is another easy win, I just don't play K+pawns v K+R very well and wanted to avoid it.

What could white have done to hold out longer?

In a losing ending, the short side needs to create counterplay, not sit around until the strong side executes the win.

35 Ra4 moves the Rook off of the 5th rank, where it had some influence.  Better is 35 a4, getting the pawn closer to the promoting rank (which could be important in some variations), and keeping the Rook where it can hinder Black to some degree.

36 hxg6 opens the door for Black. After this, White is probably lost.  In this position, it's better to let Black exchange the pawns, as the King won't be able to penetrate as easily.  White can play 36 Kb3, inching closer to a5.  If 36 ... hxg5, 37 hxg5 keeps Black's King out for now, as 37 ... f5 would be answered by 38 f4.  White will gain a few moves for other plans.

37 Ra5 is useless now, as Black has a way through with 37 ... f5.  Since the Rook has no future on the a-file, White might as well free it with 37 Rb4 so that it can at least maneuver around and allow Black to make a mistake.

38 gxf5 is playing into Black's hands.  Instead of forcing Black to make good moves, White leads Black into them.  While the position is quite lost anyway, 38 Ra4 would at least require Black to play carefully.

After this, Black has no problems.

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