Primate holding service initially in part of church (4)
I believe the answer is:
apse
'part of church' is the definition.
('apse' is part of 'church')
'primate holding service initially' is the wordplay.
'primate' becomes 'ape' (I've seen this before).
'holding' is an insertion indicator.
'initially' indicates taking the first letters.
The initial letter of 'service' is 's'.
'ape' enclosing 's' is 'APSE'.
'in' acts as a link.
(Other definitions for apse that I've seen before include "In sacred space" , "Part of a church with the altar" , "Church's part" , "Recess in church wall" , "Church area" .)