Rear half of studio, once, base for bishop's office (7)
I believe the answer is:
diocese
'bishop's office' is the definition.
(a diocese is the area a bishop is responsible for)
I cannot understand how the remainder of the clue works.
(Other definitions for diocese that I've seen before include "Bishop's see" , "Ecclesiastical see" , "District under a bishop's authority" , "I see cod (anag)" , "District under care of bishop" .)