The children of this Thomas Henry and Lydia all have very distinctive names and are thankfully easily found on the electoral rolls index.
A few of them do have a connection to Christchurch (and Canterbury and other South Island towns) at various times during their lives.
The important aspect is, this couple do not (appear) to have had a daughter by the name of Annie Kathleen ANDERSON.
As far as I can see (and using variants of the names Annie and Anderson ), there is no birth on the NZ index of an Annie Kathleen ANDERSON.
[Further, the Lydia FOSTER born 1867 is not the wife of this Thomas Henry - she would only have been 12 in 1879.]
~ Lu