White must have the winning chances, now.

In fact, this is what I would've done. Usually, I agree with ION and go by what he says because it blends in with my style. But, for a fact, I can tell you that ION is spot on here. As I used the same type of strategy to win a game against Tom Walsh in a lost position. It was posted in the WTWLTR club. Before that, I won a different game in a similar way since pawn chains are useful.

In front of it, actually... Black should probably hold this ending, but he'll have to fight for it.

This win is a little harder then I thought since I figured that white would keep the b pawn, as I did with Tom Walsh with his one mistake that turned a win into a loss, when I broke through.

Neither am I.

It is possible Black had it in mind to counter-attack on the K-side with a view to reducing the number of pawns there. Because Black has the more active rook, his chances of holding are pretty good. Now, the h-pawn's advance might have been to permit his own King to advance, OR hold back the White King from doing so. But maybe the less intuitive 27...g5 might have been the better option for holding up the White K-side.

Which is also what I'd do, keeping in mind that passed pawns must be pushed.

It was not so much the 'messing about' as the wrong 'messing about. My attitude when defending an end game is to be miserly with the pawn moves. Make the enemy weaken them. Having said that, Black might have made better progress (going back a few moves) by trying 38...f6, with the idea of forcing an exchange by ...g5. Broadly speaking, the fewer the pawns on the K-side, the fewer White's attacking prospects.

Threatens a5-a6-a7, Rh8ch, a8=Q. Another theme Black has to watch for, once the pawn reaches a7, is to avoid ...Kf7, else this will happen: 1.a7 Kf7?? 2.Rh8! Rxa7 3.Rh7ch and wins with a skewer. This is a well known motif in rook end games (I was actually on the receiving end in one of my own games on GK. Unfortunately it seems to be no longer accessible, else I would have shown it on Jo's rook endgame club thread)

Why do these games become inaccessible? I find that annoying since IM cami3 only played one game here which is inaccessible, as well.