Primate taking in second part of church (4)
I believe the answer is:
apse
'part of church' is the definition.
('apse' is part of 'church')
'primate taking in second' is the wordplay.
'primate' becomes 'ape' (I've seen this before).
'taking in' is an insertion indicator.
'second' becomes 's' (common abbreviation).
'ape' going around 's' is 'APSE'.
(Other definitions for apse that I've seen before include "part of cathedral" , "Recess in church wall" , "Church recess in lapsed condition" , "Large recess in church" , "Peas (anag) - found in church" .)