Ohh nice spot on the relationship with Henry Garthwaite, Galium
From the registers of St Mary's, Banbury
burial
15 May 1853 Henry John Garthwaite of Calthorpe Lane aged 33
baptisms
1853 May 25th George son of Henry John & Catherine Garthwaite of Calthorpe Lane French Polisher
1853 May 25th Thomsine Elizabeth daughter of Henry John & Catherine Garthwaite of Calthorpe Lane French Polisher
1853 May 25th Samuel Robert son of Henry John & Catherine Garthwaite of Calthorpe Lane French Polisher
1853 July 4th Lucy Mary daughter of Henry John & Catherine Garthwaite of Calthorpe Lane French Polisher
It looks as though Catherine had the children baptised (after the death of Henry) and registered as Garthwaite even though she and Henry John weren't married, and later they reverted to the name Bidmead I can see a likly George & Samuel Bidmead in later census (they were both Chimney sweeps). They both give their place of birth as Banbury. I think it very likly that the Lucy Mary above is the one you are looking for annevicary and that the father Samuel Bidmead on her marriage registration is made up useing her brothers name (I have seen this sort of thing before).