More on the Dunsford 'mystery'.

Some more searching has brought these facts to light. Henry Dunsford was a Sea Captain, so rarely if ever shows up in the Census. However, his wife and family can be found at Honiton Clyst in 1851, with information that suggests that they returned from Sydney to Devon in 1849, with a daughter, Fanny, being born At Sea on the way home.

1851, Clist Honiton (sic), Devon HO 107/1866 f.158 p.14

Vernon House
Mary Dunsford, Head, married, 32, Captain's wife, born Devon, Clist Honiton (sic)
Henry Dunsford, son, 6, born Sidey (sic), Australia
Sophia Dunsford, daughter, 4, Scholar, born do do
Fanny Dunsford, daughter, 2, Scholar, born do do
Henry Fish, relative, unmarried, 23, Cordwainer, born Devon, Clist Honiton (sic).

1861, Heavitree, Exeter
Fanny Dunsford, Servant, unmarried, 12, Servant, born At Sea. (transcribed as Sea, Devon) (This makes more sense than Sydney, because she's not in the NSW BDM that I can see.). Fanny married William Underwood.

Only possible marriage I can see for the parents is Henry Dunsford to possibly Mary Petter, Barnstaple, Dec Q 1841, Vol. 10, p. 34

The Honiton Clyst location might suggest that either Henry or Mary Dunsford were friends of the White family. I note in Jack Earnshaw's tree that there is a marriage of an Elizabeth Dunsford to Joseph Wheaton in 1800 in Otterton, and their daughter Sarah married James Bridle, brother of Susan Bridle who was, of course, TB White's mother. So there is a clear relationship between some Dunsford names and the White family.