Part of church where service initially is covered by primate (4)
I believe the answer is:
apse
'part of church' is the definition.
('apse' is part of 'church')
'service initially is covered by primate' is the wordplay.
'initially' suggests taking the first letters.
'is covered by' is an insertion indicator.
'primate' becomes 'ape' (I've seen this before).
The initial letter of 'service' is 's'.
's' put into 'ape' is 'APSE'.
'where' is the link.
(Other definitions for apse that I've seen before include "Recess at east end of church" , "Church recess in lapsed condition" , "Offshoot from the choir, nave or aisle in a church" , "inside church?" , "Church area" .)