Singer has liberal shrink (6)
I believe the answer is:
flinch
'shrink' is the definition.
(I know that shrink can be written as flinch)
'singer has liberal' is the wordplay.
'singer' becomes 'finch' (finch is a kind of birdanimal).
'has' is an insertion indicator.
'liberal' becomes 'l' (abbreviation).
'finch' going around 'l' is 'FLINCH'.
(Other definitions for flinch that I've seen before include "Cower, baulk" , "Wince or draw back in fear" , "Draw back in fear, recoil" , "Start back in pain or shock" , "Recoil from pain, surprise etc" .)