Leave course having lost heart (6)
I believe the answer is:
desert
'leave' is the definition.
(desert can mean to leave or abandon)
'course having lost heart' is the wordplay.
'course' becomes 'dessert' (dessert is a kind of course).
'having lost heart' suggests removing the centre.
'dessert' with its centre removed is 'DESERT'.
(Other definitions for desert that I've seen before include "Hot dry area" , "See 23" , "Arid area of sand" , "Walk out on" , "land too dry to raise crops" .)