Hi Harriet
Both the Otago Benevolent Institution and the Caversham Industrial School were operating from the late 1860's.
This was in the middle of the Otago goldrush.
Unfortunately the Registers of inmates at the Benevolent are
1887-1897
1898-1914
Also, the Outdoor relief casebooks date from 1889.

There were Rough minutes of the Otago Benevolent Institution, but I didnt think they would cover individual cases.
The several volumes relating to the Caversham industrial school
are also a later period, and held in Christchurch.
Have you looked to see if the father remarried?
BerylM