The following Jerome Gambit game is a good example of what to look at.
Wall, Bill - Guest129367
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+
4...Kxf7 5.Nxe5+ Kf8
Checking Bill's Jerome Gambit nomenclature, I am reminded that this is the Sorensen Variation, where Black declines the second sacrifice and moves his King closer to safety.
Lieutenant S. A. Sorensen wrote an early analysis of the Jerome Gambit in his "Chess for Beginners" column in the May 1877 issue of Nordisk Skaktidende. It was widely translated and republished.
He was anticipated in his discussion by Alonzo Wheeler Jerome, who first looked at the line in 1874. The earliest game example that I have in The Database is Jerome - Brownson, Iowa, USA, 1875 (1/2-1/2, 29). For history of the line, see "Critical Line: 5...Kf8 (1, 2 and 3)".
I was a bit surprised to find that, according to The Database, this move is a novelty, although the game will transpose to an earlier Bill Wall game in a few moves.
White simply develops a piece, and waits to see what Black can make of the position. Similar would be 6.0-0, which Bill is 7-0 with. He also suggests 6.d4!?, which brought him a win the one time he tried it.
Seen more often, and recommended by both Jerome and Sorensen, is 6.Nxc6.
The Black Queen sometimes goes to f6 to help defend; here it is also attacking.
Of course Black had the option of capturing the Knight with 6...Nxe5, and after 7.d4 Bxd4 8.Qxd4 d6 the game would have transposed into more mainstream play. There is a practical argument for 6...Nxe5 as well: Bill is "only" 8-2 against it.
Stockfish 8 (30 ply) shows a tiny preference for 6...Nxe5, but also suggests the wild 6...Qg5!?, which it recommends that White meet with 7.Qf3+, looking to exchange Queens. Not surprisingly, The Database has no examples of 6...Qg5.
White can no longer exchange off his Knight because of 7.Nxc6?? Qxf2 checkmate.
7...Bb6 8.O-O d6
The game has transposed to Wall, Bill - Tim93612, Chess.com, 2010 (1-0, ), which began 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+ Kxf7 5.Nxe5+ Kf8 6.Nd3 Bb6 7.0-0 Qf6 8.Nc3, but then continued with 8...Nge7 (1-0, 36).
With a piece for two pawns, Black can be said to have a small advantage, especially in light of White's blocked development. Black has to be aware, however, that his King and Queen are on the same file, which could prove risky if White can exchange off pawns to expose his Rook at f1.
White wastes no time unpinning his f-pawn, so he can advance it.
Black would love to exchange Knights at d3 and bury White's Bishop. Bill recommends, instead, 9...Qf7.
The attack on Black's Queen gives White time to exchange off Black's troublesome Bishop and double a couple of pawns - if nothing better comes up.
Why didn't White play the Knight jump the previous move, instead of "wasting time" with 9.Kh1 ? Probably because Black could have answered the move with 9...Qd4, and the Queen would have recaptured, keeping his pawns intact.
Bill suggests, instead, 10...Qe6 11.N3f4 Qf7. Why is this different from moving the Queen to f7 immediately?
11.Nxe5 dxe5 12.f4
Suddenly the "Jerome pawns" take on a menacing potential.
Stockfish 8 suggests that Black continue with reasonable defense, 12...Ke8 13.fxe5 Qg6 14.d4 Ne7 15.Nxe7 Kxe7 16.Qd3 and with 3 pawns for the piece - despite facing the two Bishops - White appears to have somewhat better chances.
Black's first real slip. Development is good, and shielding the Queen and King is noble - but the enemy Rook on the f-file is still a danger.
13.fxe5 c6 14.exf6 g6 15.Ne7
Not only is White up 3 pawns, his advanced Knight and pawn are full of trouble - especially with Black's pawn on g6. (Perhaps Black's Queen should have gone there, instead, with 14...Qg6, but after 15.Nxb6 axb6 White could have continued with 16.b3, intending 17.Bb2 with further pressure on Black's kingside.)
15...Be6 16.d4 Rd8
Overlooking White's main threat, which could have been met with 16...h5. At that point White would not have an immediate shot, but could continue to build his attack with 17.b3 and 18.Ba3, or work for a breakthrough with 17.d5.
Now there is already the win of the exchange after 18.Bg7, but White wants to go after the enemy King.
It is difficult to find a safe retreat for Black's light squared Bishop, so Black decides to return it for a couple of pawns. This leaves him down a Rook (and a couple of pawns)
19.exd5 Bxd5 20.Nxd5 Rxd5
Please excuse Bill for overlooking the checkmate in 20 moves that starts with 21.Qg4 (silly computer) - the text is strong enough (and would checkmate almost as fast, if Black didn't resign sooner).
21...Kd7 22.Qg4+ Kc7 23.Re7+ Black resigned