You may already have seen the birth register entry:
BROWNE, Anna Edith, born 29 December 1857
Father: Browne, James
Mother: Gooch, Ellen
Registered: Hobart, 1858,
Record ID:
NAME_INDEXES:962706
Resource: RGD33/1/7 no 1188
The entry shows that the mother Ellen BROWNE, formerly GOOCH, Church St, is the informant. Father's occupation: listed as a mechanic. The register has no name for the child but added later "Baptised St Davids Anna Edith"
https://linctas.ent.sirsidynix.net.au/client/en_AU/names/Do you know where Ellen was with Sir Valentine FLEMING? There is an Ellen GOOCH having a child registered Launceston, 1854. The father of the child is listed as John FIELD, farmer and there is a notation saying that the parents are not married. However, on looking at information about Valentine FLEMING he seems to have been in Hobart, at least for his marriage 1852 and the births of his children in the early 1850s so I think this is probably not your person.
Judith