Become discouraged (4,5)
I believe the answer is:
lose heart
'become discouraged' is the definition.
(I've seen this in another clue)
This is all the clue.
(Other definitions for lose heart that I've seen before include "Become discouraged in cardiac manner?" , "What patient may do before subsequent transplant" , "Let hopes dwindle - real ethos involved" , "find discouragement" , "Falter in courage - or fall in love? (4, 5)" .)