Checking the 1881 census, where you can do a search by streetname - Clare Street appears in District 23 of Islington, St Anne's Ecclesiastical District. The Enumerator's Walk reads
Part of the City of Liverpool comprising Springfield Street North Side from St Anne Street to Christian Street. Clare Street both sides and Courts from Springfield Street to Myrtle View. St Anne Street West Side and Courts from Springfield Street to Great Richmond Street.
If you check the current maps you would see Springfield Street on the east of St Anne's Street, but I would think that Clare Street, with Back Clare Street, would be to the West of it, between there and Christian Street.