Churchwarden say quietly slices pie (4)
I believe the answer is:
pipe
'churchwarden' is the definition.
(I've seen this in other clues)
'quietly slices pie' is the wordplay.
'quietly' becomes 'p' (p for piano and quiet).
'slices' is an insertion indicator (inserted letters slice their way in).
'p' going into 'pie' is 'PIPE'.
'say' acts as a link.
(Other definitions for pipe that I've seen before include "Smoker's item" , "Long tube" , "Part of an organ; tube" , "Tobacco is smoked in it" , "Tube; large cask" .)