Cut a part of the nose (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'
(Other definitions for abridge that I've seen before include "Shorten (eg, text)" , "Make shorter, book say" , "Shorten written work" , "Abbreviate, shorten" , "(US) cattle-pen" .)