For what it's worth, let's look at some numbers, based on my Jerome Gambit (1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+) database.
Please keep in mind the caveats about this that I mentioned in my reply to Pete Bank's ("blackburne") Comment to "Fool me once..."
I use ChessBase 8 to store most of my Jerome Gambit games and findings.
If I look at the series of moves 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+, and run an Opening Report, I can see that ChessBase finds 1,436 relevant games. About these, it notes that White scores above average – 57%; that is, White wins 693 (55%), Draws 32 (3%), and Black wins 528 (42%). Not surprisingly, it points out that the Drawing number is quite low, with 1/3 of those drawn games having fewer than 20 moves.
In addition, White's wins are "shorter than average" (35 moves), Black's wins are "short" (29 moves), and the Draws are "shorter than average" (34 moves).
Looking at Moves and Plans, I can see that 1217 games featured 4...Kxf7, with which Black scored "averagely" – 44%. The Opening Report recommends 5.Nxe5+ for White, noting that 5...Nxe5 6.Qh5+ Ke6 7.f4 d6 then occured in 97 games; 5...Nxe5 6.Qh5+ Ng6 occurred in 118 games; and 5...Nxe5 6.d4 Qh5 7.0-0 Nf6 8.dxc5 Qxe4 9.Nc3 Qb4 10.Nd5 occurred in 118 games. That last line is highly skewed: all 118 games were from a match between Deep Shredder 10 and Rybka 2.3.1.
Interestingly enough, the "Critical line" for White, in which he scored only 29%, is 5.Nxe5+ Kf8 6.Nxc6 dxc6, which certainly bears some looking into.
Althought taking the sacrificed Bishop seems the thing to do, the Opening Report shows that in 19 games Black played 4...Kf8 instead, scoring "below average" at 37%. The recommended response for White is 5.d3.
In 5 games Black responded with 4...Ke7, when he "scored miserably" with 20%. White's response should be 5.Bh5.
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