Support taking depressed chicken out? (4,4)
I believe the answer is:
back down
'chicken out?' is the definition.
('back down' can be a synonym of 'chicken out')
'support taking depressed' is the wordplay.
'support' becomes 'back' (back is a kind of support**).
'taking' means one lot of letters go next to another.
'depressed' becomes 'down' ('down' can be a synonym of 'depressed').
'back'+'down'='BACK DOWN'
(Other definitions for back down that I've seen before include "Withdraw argument or avoid confrontation" , "Withdraw argument in face of opposition" , "Give up claim or withdraw argument" , "Withdraw opinion or claim" , "admit defeat" .)