Fail to win compassion, become despondent (4,5)
I believe the answer is:
lose heart
'become despondent' is the definition.
(I've seen this before)
'fail to win compassion' is the wordplay.
'fail to win' becomes 'lose' ('lose' can be a synonym of 'fail to win').
'compassion' becomes 'heart' (I've seen this before).
'lose'+'heart'='LOSE HEART'
(Other definitions for lose heart that I've seen before include "Falter in courage - or fall in love? (4, 5)" , "What patient may do before subsequent transplant" , "Start to despair" , "Become despondent" , "find discouragement" .)