Hang on - this is fascinating!
I am going to contend that it is won for Black (which makes the earlier Kg5 a very, very bad move), but there are a lot of variations. I have worked these all out myself with a board and the PGNs here over the last two hours, so there may yet be a hole in the analysis.
First, White has an extra pawn he can use and thinks he has a win:
[FEN "7K/8/8/2p5/1pP5/8/PP2k3/8 w - - 0 1"]
1. a4 {Trying to create a passed pawn as quickly as possible, but...} 1. ... bxa3 {Taking <i>en passant</i> here is, of course, obligatory - unless Black wants to allow the a-pawn to promote!} 2. bxa3 Kd3 3. a4 Kxc4 4. a5 Kb5 {... Black's King is in the square, and can pick off a crucial pawn on the way; the c5 pawn promotes}
So White must bring the King over, as greenpawn suggested, but can Black attack the two back pawns and win by promoting the b-pawn more quickly?:
[FEN "7K/8/8/2p5/1pP5/8/PP2k3/8 w - - 0 1"]
1. Kg7 Kd2 2. Kf6 Kc2 3. Ke5 Kxb2 4. Kd5 Kxa2 5. Kxc5 b3 6. Kd6 b2 7. c5 b1=Q {"And Black wins" - though it isn't so simple to the uninitiated, and I spent a little time working it out. The winning method is:} 8. c6 Qb5 9. Kd7 Qd5+ 10. Kc7 Kb3 11. Kb6 Qd4+ 12. Ka6 Qc5 13. Kb7 Qb5+ 14. Kc7 Kc4 15. Kd6 Qd5+ 16. Kc7 Kc5 {And the pawn falls, leaving a simple mate}
I've repeatedly been through the a-pawn attempt we saw earlier, tried once the Black King is closer, and I don't think there's a draw for White. First, the race:
[FEN "7K/8/8/2p5/1pP5/8/PP2k3/8 w - - 0 1"]
1. Kg7 Kd2 2. Kf6 Kc2 3. a4 bxa3 4. bxa3 Kb3 5. Ke5 Kxc4 6. a4 Kb4 7. Kd5 c4 8. a5 c3 9. a6 c2 10. a7 c1=Q 11. a8=Q {Here we are again!} 12. Qh1+ {.... and another skewer wins for Black}
What about moving the King out of the way of that check, then advancing the a-pawn?
[FEN "7K/8/8/2p5/1pP5/8/PP2k3/8 w - - 0 1"]
1. Kg7 Kd2 2. Kf6 Kc2 3. a4 bxa3 4. bxa3 Kb3 5. Ke5 Kxc4 6. a4 Kb4 7. Kd5 c4 8. Kd4 c3 9. a5 c2 10. a6 c1=Q 11. a7 {11 ... Qa1 or 11 ... Qa3 or 11 ... Qc8 or 11 ... Qc6 all stop the pawn, but I think most aesthetically pleasing is...} 11. ... Qg1+ {... with a similar skewer again}
So can White just stay with the c-pawn and prevent it promoting?
[FEN "7K/8/8/2p5/1pP5/8/PP2k3/8 w - - 0 1"]
1. Kg7 Kd2 2. Kf6 Kc2 3. a4 bxa3 4. bxa3 Kb3 5. Ke5 Kxc4 6. a4 Kb4 7. Kd5 c4 8. Kd4 c3 9. Kd3 Kb3 {Of course, if Black takes the a-pawn, White takes the c-pawn with an easy draw} 10. a5 c2 11. a6 {Or 11. Kd2 Kb2 12. a6 c1=Q+ and then scooping up the a-pawn wins} 11. ... c1=Q 12. a7 Qc8 {Stopping the a-pawn, and winning}
So, White can't draw with the a-pawn attack (unless my fatigued mind has missed something?). What about delaying Black by defending the b-pawn?
[FEN "7K/8/8/2p5/1pP5/8/PP2k3/8 w - - 0 1"]
1. Kg7 Kd2 2. Kf6 Kc2 3. b3 Kb2 4. Ke5 Kxa2 5. Kd5 Kxb3 6. Kxc5 Ka4 7. Kd6 b3 8. c5 b2 9. c6 b1=Q 10. c7 {And K+Q vs K+BP on 7th is a well-known draw - usually!} 10. ... Qb7 11. Kd7 {aiming for Kd8, c8 and a draw} 11. ... Kb5 12. Kd8 Qd5+ 13. Ke8 {The Queening square is no longer attacked by the Black Queen, so the King can't advance...} 13. ... Qc6+ 14. Kd8 Qd6+ 15. Kc8 {Can't go to e8 this time, as the c-pawn is attacked} 15. ... Kb6 16. Kb8 Qxc7+ 17. Ka8 Qb7 {And that is mate}
I have no other ideas for White - anyone?