A bit late in the day as far as replies to this thread are concerned, but Handsworth Historical Society's (HHS) information on The Moors from our 1898 directory shows it as 12 Church Lane (then occupied by Mrs Sarah Eliza Goode) and it was at the Grove Lane end of Church Lane. There was also an unnumbered house called Moorfields almost opposite, next to No. 13, occupied by O H Duffell. At that time, the directory shows only six houses on The Moors side, with fields in between the 3rd and 5th, and ten houses on the other side.
It housed Belgian refugees during WW1 and later became a nursing/maternity home before demolition. There's a painting of The Moors on Birmingham Images and HHS also has photos of it including a brochure of the maternity home. Some 2002 correspondence from descendants of the Goode family said that they moved to Selborne Road in 1926 as their mother was a widow, and several members of the family are buried at St Mary's Handsworth.
Hope this helps.