In the Jerome Gambit (1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+), White often finds himself with one, two, or even three extra pawns for the piece that he exchanges.
I suspect that stronger players and chess computer engines would prefer to have the piece, and would see the pawns as targets, especially in the middle game.
On the other hand, I think many club players would prefer the pawns, especially in the middle game - where the "Jerome pawns" can advance ahead of an attack - or in the endgame - where the clock is ticking, and it can be easier to "push a pawn" than position a piece.
The following game is worth playing over several times, to see how White champions the pawns, and how they lead him on to victory.
Wall, Bill - Guest1590349
PlayChess.com, 2020
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+
Bill is 9 - 0 from this standard position, which goes back at least as far as Charlick - Mann, corresponcence, Australia, 1881 (1-0, 72); and is as recent as
10.b3 Kf7 11.O-O Re8 12.d3 d5 13.f3 dxe4 14.fxe4 Qd6 15.Bb2 Kg8
Black and White both have made progress. Black has castled-by-hand and executed the pawn break ...d5, opening lines for development. White has maintained his solid, if constrained, pawn center and placed his Bishop on the open diagonal, pointing at the Knight at f6, which is also under pressure from his Rook at f1.
Stockfish 11 gives Black an advantage - less than a pawn - consistent with having a piece for two pawns.
16.h3 Nd5
Taking advantage of the pin on the e-pawn to attack the Queen, but nothing comes of it. It was probably better to just develop with 16...Be6.
17.Qd4 Nf6 18.Qe3 Nd5 19.Qd4 Nf6 20.Qxd6 cxd6 21.Bxf6 gxf6 22.Rxf6 Re6
White has created weaknesses in his opponent's position. Stockfish 11 still gives Black a small advantage, but I think many Jerome Gambit players would be comfortable with 3 pawns vs the piece. It is good to watch Bill work with what he has.
23.Rf3 Bd7 24.Nd2 Rf8 25.Raf1 Rxf3 26.Rxf3 Kg7 27.Kf2 Rf6
Black is happy to pin the enemy Rook to the King, allowing another exchange (When ahead in material, exchange pieces). He underestimates the activity of the enemy Knight.
28.Nc4 d5
Getting to play ...d5 twice in a double e-pawn game had to feel great. Fearing the loss of his pawn, Black gives it up, hoping to disrupt White's pawn structure.
29.exd5 b5 30.Nd2
30...Ne5
Misjudging the position. His Rook is worth preserving, to provide counterplay on the Queen's wing, e.g. 30...Ra6 31.a4 bxa4 32.bxa4 Rxa4 33.c4 Ra2 34. Ke3 a5. With a passed a-pawn, if his King can blockade White's central pawn mass, he might be able to control things to keep the game even.
31.Rxf6 Kxf6 32. d4 Ng6 33. c4 bxc4 34. bxc4 Ne7
35.Ne4+ Kf5 36.Ke3 Nc8 37.g4+ Kg6 38.c5 Kf7
Can we take a moment to remember the "Killer Shrews"?
39.d6 Ke6 40.Ng5+ Kd5 41.Nxh7 a5 42.Nf6+ Ke6 43.Nxd7 Kxd7 44.g5 Black resigned
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