Primate crossing small part of church (4)
I believe the answer is:
apse
'part of church' is the definition.
(part of a church building)
'primate crossing small' is the wordplay.
'primate' becomes 'ape' (ape is a kind of primate).
'crossing' is an insertion indicator.
'small' becomes 's' (abbreviation - e.g. clothes size).
'ape' going around 's' is 'APSE'.
(Other definitions for apse that I've seen before include "part of cathedral" , "Recess with domed rof in a church" , "may be seen in church!" , "where praetor sat" , "Church recess in lapsed condition" .)