For the benefit of anyone following this thread, I now have the birth certificate of Georgina Ann Mole.
Father - George Mole , Brassfinisher
Mother - Caroline Mole nee Ockleford.
So it appears to me that Caroline had an illegitimate child while her husband Henry Walkinshaw was away at sea.
Georgina is a Mole in civil records. By the time of christening in 1859 she is an Ockleford and she is a Walkinshaw in the 1861 census. In 1866 she is re-christened as a Walkinshaw and any reference to George is dropped from her name, leaving her as Annie. Caroline and Henry went on to have many other children together from 1861.
In the Directory of Hampshire and Isle of Wight for 1859 in Southampton there are no George Moles. Nor any in directories before or after. No one is listed at the Cambridge Street address either in 1859.
So I now have to look for him elsewhere -and see if he is a real person.