Thanks ShaunJ. This location would make sense. I managed to find a map (Philip's plan of the town and port of Liverpool with Birkenhead and the adjoining Cheshire coast, Bartholomew, John London, [1884?]) and although I didn't find Stanley Square on it, there are unnamed alleyways or streets off Dryden St close to the eastern (Great Homer Street) end to the south and also to the north. Perhaps one or other of those forms the square.
Still no luck with Cavendish Road.