Hello,Eillen,
I have checked White Cross Street in the 1851 census & cannot find a Clayhouse Yard but there is a Playhouse Yard, probably a mistranscrirption somewhere.
1851 census
Playhouse Yard White Cross Street St Luke
HO105/1522 folios 175-180
145-150
I checked thro' these folios but could not identify a school for the homeless, the occupants all seemed to be general workers & traders.
Hope this helps,
Meliora