An addition adjoining eastern part of church (4)
I believe the answer is:
apse
'part of church' is the definition.
('apse' is part of 'church')
'an addition adjoining eastern' is the wordplay.
'an' becomes 'a'.
'addition' becomes 'ps'.
'adjoining' is a charade indicator (letters next to each other).
'eastern' becomes 'e'.
'a'+'ps'+'e'='APSE'
(Other definitions for apse that I've seen before include "Church recess in lapsed condition" , "Church recess with domed or arched roof" , "Where you could find a magistrate's chair" , "where praetor sat" , "Recess at eastern end of a church" .)