Family Search has a transcription of the marriage on 19 March 1920 at St Peter's, Birkenhead. Robert Lawrence SWAIN is 30 and his father is William Henry SWAIN (no addresses, witnesses or occupations given in the transcription).

One possibility is Robert Lawrence SWAIN born 30 Dec 1889/1890 in Stanton/Stranton, Virginia USA (this could be Staunton).

Passenger list:
22 Feb 1921 SS "Atenas" Cristobal C.Z. to New Orleans La.
Consular passenger (destitute American seaman) placed aboard by US Shipping Commissioner at Cristobal C.Z.
SWAIN Robert L., 31, male, married, born Stanton, Virginia 30 Dec 1889, Left at hospital by Amer. S/SWest Notus 9 Jan 1921, Address in USA 1115 Prytania St, New Orleans, La.

The Independent Record from Helena, Montana, page 3, Saturday 22 Feb 1936:
Article about missing veterans sought by the Veterans Administration:
"Robert Lawrence SWAIN disappeared from his home in Boston Mass. December 1922, born 30 Dec 1890 Stranton Va, served at Cook Co. Com Center 101st Depot Brigade Washington

Of course this could be a complete red herring! It would be useful to know the occupation and address of your Robert Lawrence when he married. It might be possible to track him through electoral registers. Or if he was a US sailor, perhaps he met Minnie on a trip to the UK, then abandoned her after the marriage. As the wedding was on 19 March and Robert Henry was born in December quarter of that year, it is quite possible that Minnie was already pregnant when she married. Robert L. may not have been the father after all and that is why he left. Robert H. may have been put into care rather than being taken away by his father.