Priest entered a southeastern part of church (4)
I believe the answer is:
apse
'part of church' is the definition.
('apse' is part of 'church')
'priest entered a southeastern' is the wordplay.
'priest' becomes 'p' (ecclesiastical abbreviation).
'entered' is an insertion indicator.
'southeastern' becomes 'se'.
'a'+'se'='ase'
'p' inserted into 'ase' is 'APSE'.
(Other definitions for apse that I've seen before include "Semicircular end of a church, behind the altar" , "Recess in church wall" , "Recess at east end of church" , "In sacred space" , "Niche in church" .)