Last Greek put out by the church (4)
I believe the answer is:
kirk
'the church' is the definition.
(kirk is a kind of church)
'last greek put out' is the wordplay.
'last' suggests the final letters.
'put' means one lot of letters go next to another.
'out' becomes 'irk' (I can't justify this - if you can you should believe this answer much more).
The last letter of 'greek' is 'k'.
'k'+'irk'='KIRK'
'by' acts as a link.
(Other definitions for kirk that I've seen before include "Scots church" , "Star Trek captain - Scottish church" , "Presbyterian building?" , "Church of Scotland" .)