First, this is not really a Polerio, aka Muzio Gambit (this is the only name I have ever heard it called). That is characterized by the move 3.Nf3 g5 4.Bc4 g4 5.0-0.
My first impression is that I see a lot of tactical mistakes. There is a beautiful tactic that was missed on move 13. Here are my notes to the game.
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1. e4 e5 2. f4 exf4 3. d4 $4 {This is almost losing already.} (3. Nf3 $1 g5 4. Bc4 g4 5. O-O {This is the Muzio/Polerio.}) 3... Nf6 $2 (3... Qh4+ {This is close to winning already.} 4. Ke2 {This is forced, and if so, white needs to deviate sooner.} (4. g3 $2 fxg3 5. Nf3 g2+ $1 6. Nxh4 gxh1=Q $19)) 4. Qf3 $6 Bb4+ $2 {This allows white to solidify the center with tempo.} (4... Nc6 $1 5. c3 d5 6. e5 Ne4 $17 {And black is better.}) 5. c3 Ba5 $2 {The bishop is really needed for the defense of the king.} (5... Be7 $1 {But white is still much better after e5.}) 6. Bxf4 $2 (6. e5 {This drives the knight back to its home, and it clearly the best since white will be way ahead in development.} Ng8 7. Bxf4 Ne7 8. Bd3 {With a huge advantage.}) 6... O-O 7. Nh3 $4 (7. e5 $1) 7... Re8 $4 (7... Nxe4 $1 {This just wins a pawn, and black went from much worse to at least equal.}) 8. e5 d6 $1 {The white king is now stuck in the middle, so black will be winning material.} 9. Bb5 (9. Nd2 c6 10. Bc4 dxe5 11. Bxf7+ Kxf7 12. Ng5+ Kg8 13. Bxe5 Nfd7 $4 14. Qf7+) 9... c6 (9... Bg4 10. Qd3 Nbd7 {Finishing development is even better.}) 10. Bc4 dxe5 11. Bxf7+ {This is nothing more than a check. Black can take this with impunity.} Kxf7 12. Ng5+ Kg8 13. Bxe5 Nfd7 $4 {Black is crushing here, but instead walks into mate in two. Almost any move here wins, but this one. That said, black had a beautiful, and unique tactic here that wins another piece.} (13... Qxd4 $1 {Exploiting two pins, this wins the Be5.}) 14. Qf7+ 1-0