Leave course, disheartened (6)
I believe the answer is:
desert
'leave' is the definition.
(deserting is a kind of leaving)
'course disheartened' is the wordplay.
'course' becomes 'dessert' (dessert is a kind of course).
'disheartened' suggests removing the centre.
'dessert' with its centre taken out is 'DESERT'.
(Other definitions for desert that I've seen before include "Abandon in barren terrain" , "Uncultivated" , "Callously abandon" , "Sahara or Gobi, say" , "Leave behind - waste" .)