Hi,
Welcome to RootsChat

If I'm looking at the right family, I can see Emily DAVIS age 67 with her daughter Emily age 37 living in Elliott Place in 1851.
The description of the enumeration district says,
All that part of the Parish of Alverstoke which is comprised in Bingham or in New Town including,
Richard Street
York Terrace
Henry Street
Charles Street
Joseph Street
St John's Place
Elliott "
The houses in Stoke Road
Grove Field
Grove Buildings and
Chester Place
Holly St
On this map I can find some of these streets etc. in Newtown (Just below the large Gosport word).
https://maps.nls.uk/view/102343170I found an old town plan of the area,
https://maps.nls.uk/view/229916130Which shows Elliott Place to be some houses at the end of Stoke Road.

Here is a side-by-side view,
https://www.rootschat.com/links/01r5w/and looking at G... Street view I think it's possible that the houses might still be there. If so that would be a really lucky find in Gosposrt as most of it has been rebuilt.
Regards,
Daisy