Becomes bloody disheartened, but copes (4,2)
I believe the answer is:
gets by
'copes' is the definition.
('get by' can be a synonym of 'cope')
'becomes bloody disheartened' is the wordplay.
'becomes' becomes 'gets' ('get' can be a synonym of 'become').
'disheartened' says to hollow out the word (remove centre letters).
'bloody' with its middle removed is 'by'.
'gets'+'by'='GETS BY'
'but' acts as a link.
(Other definitions for gets by that I've seen before include "Manages" , "makes do" , "copes" , "Planet Herschel found" .)