Communist at posh church cut down (6)
I believe the answer is:
reduce
'cut down' is the definition.
(I know that cut down can be written as reduce)
'communist at posh church' is the wordplay.
'communist' becomes 'red' (red can informally mean communist).
'at' is a charade indicator (letters next to each other).
'posh' becomes 'u' (abbreviation for upper-class - opposite to non-U).
'church' becomes 'CE' (abbreviation for Church of England).
'red'+'u'+'ce'='REDUCE'
(Other definitions for reduce that I've seen before include "Slim down or discount" , "Pare" , "Decrease" , "Make smaller or take off weight" , "Lower" .)