Live recording disheartened composer (4)
I believe the answer is:
berg
'composer' is the definition.
(20th Century Austrian composer)
'live recording disheartened' is the wordplay.
'live' becomes 'be' (synonyms).
'disheartened' says to hollow out the word (remove centre letters).
'recording' with its middle removed is 'rg'.
'be'+'rg'='BERG'
(Other definitions for berg that I've seen before include "Viennese composer" , "African mount" , "Alban ----, composer of the incomplete opera 'Lulu' (1935)" , "He composed" , "Alban -, composer (Wozzeck, Lulu)" .)