I feel sure that John's first guess is correct.
It is not a DH.34B which was a single engine aircraft.
Assuming that it had a CAA registration, three Handley Page HP W8B's are listed in the register.
G-EBBG dates from 1924 to 1928
G-EBBH dates from 1924 to 1931
G-EBBI dates from 1926 to 1932.
It appears not to be the three engined variant and therefore has to a B, C, F or G and B is the most likely. The diffferences relate to the engine type.
The W10 was also a twin engined aircraft.
Unfortunately I am unable to find any evidence of this aircraft type flying the Atlantic and I am doubtful whether it had the range; it was mostly used for short flights into Europe, typically London/Paris.
Not too helpful I'm afraid!
I would be inclined to contact the Royal Air Force Museum at Hendon.
This is an example from their History of Aviation time line and relates to 1919 and includes civil flights. Doubtless they have similar info for later dates.
http://www.rafmuseum.org.uk/research/history-of-aviation-timeline/interactive-aviation-timeline/british-civil-aviation/1919.aspx
Malcolm
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