This is not a Staunton gambit.
In the Staunton Black captures with the f-pawn. Here he has captured with the d-pawn.
As Mnb states, this is a very important difference, so I have raised it here in this forum as I believe that it is essentially a BDG line not a Dutch one.
From what I can see there are 4 main move for white on move 4.
f3
Bf4
Bg5
Bc4
f3 is a mistake. As Mnb has shown 4.... e5 is very strong here.
Williams and Pinski give Bf4 (to stop e5) as their main line.
Williams gives :-
4. Bf4 Nf6 5. f3 e6 6. fxe4 (6. Bc4 ? Nd5) fxe4 7. Bc4 Nc6
8. Nge2 Na5! 9. Bb3 Nxb3 10. axb3 Bd7 11. Ng3 Bc6 12. 0-0 and now 12 ...Be7 ?! was Benjamin - Malaniuk
- but Williams give 12....Bd6! as =/+ (all annotation is Williams' not mine).
I haven't got Pinski in front of me at the moment, I will check tonight.
Sawyer in BDG Keybook 2 gives 4. Bg5 as his main line. Here 4....g6 (intending Bg7 and Nc6 putting pressure on d4 ) looks interesting, although he does give several other lines.
4.Bc4 is Pocket Fritz's favourite move - it won't play anything else. I haven't gone to deep into it but there are many transpositions into the Bg5 and Bf4 lines.
Esio