Primate going round southern part of church (4)
I believe the answer is:
apse
'part of church' is the definition.
(part of a church building)
'primate going round southern' is the wordplay.
'primate' becomes 'ape' (large primate).
'going round' indicates putting letters inside (some letters go around others).
'southern' becomes 's' (abbreviation).
'ape' enclosing 's' is 'APSE'.
(Other definitions for apse that I've seen before include "Recess in church wall" , "Part of a church with the altar" , "Church's part" , "Primate holds saint to be a" , "Where you could find a magistrate's chair" .)