There is an arrival record for Allen and Olive in New York on 14th April 1920 but the entry has been crossed out.
It gives Allen's mother-in-law as Mrs. Gillett, 128 Harrowby Road, Grantham. It says that they are heading for Lowell, Mass.
Children are Dorothy and Gordon.
There is a second entry on 16th May, giving Allen's father-in-law as Mr. J Gillett, 128 Harrowby Road. Allen is a motor engineer. Olive is shown as Clare but the previous record is Clara - Olive Clara Gillett, born in Peterborough.
Allen arrived back in Liverpool in April 1932. He is a chauffeur of 56 High Street, Grantham.
Olive and the children arrived back in July but their destination is Cadby (Cadeby?), Leicestershire.
In 1930, Allen is a garage mechanic, living in First Street, Tewksbury, Massachusetts, with Olive and the children.
As Geoff suggests, it sounds as though they were renting above a shop, given Allen's occupation.
David
P.S. I have since noticed that there is a tree on Ancestry with photographs of Allen and his parents.