Composer wearing navy skirts (10)
I believe the answer is:
stravinsky
'composer' is the definition.
(I know this)
'wearing navy skirts' is the wordplay.
'wearing' is an anagram indicator (wear can mean to become damaged).
'navy'+'skirts'='navyskirts'
'navyskirts' with letters rearranged gives 'STRAVINSKY'.
(Other definitions for stravinsky that I've seen before include "Russian composer" , "Igor -, Rite of Spring composer" , "Scorer" , "he noted" , "Igor __, "Firebird" composer" .)