A part of the nose is cut (7)
I believe the answer is:
abridge
'cut' is the definition.
('abridge' can be a synonym of 'cut')
'a part of the nose' is the wordplay.
'part of the nose' becomes 'bridge' ('bridge' is part of 'nose').
'a'+'bridge'='ABRIDGE'
'is' acts as a link.
(Other definitions for abridge that I've seen before include "Make a precis of" , "Curtail" , "trim" , "Shorten written work" , "Big read (anag) - shorten text" .)