Primate touring southern part of church (4)
I believe the answer is:
apse
'part of church' is the definition.
('apse' is part of 'church')
'primate touring southern' is the wordplay.
'primate' becomes 'ape' (I've seen this before).
'touring' means one lot of letters goes inside another (touring can mean going round).
'southern' becomes 's' (abbreviation).
'ape' enclosing 's' is 'APSE'.
(Other definitions for apse that I've seen before include "Part of a church with the altar" , "Recess with domed rof in a church" , "Where the altar is usually found in a church" , "Projection to the east end of a church" , "Church recess with domed or arched roof" .)