Count is really disheartened (4)
I believe the answer is:
rely
'count' is the definition.
(I've seen this before)
'really disheartened' is the wordplay.
'disheartened' suggests removing the centre (the middle or heart is removed).
'really' with its centre taken out is 'RELY'.
'is' acts as a link.
(Other definitions for rely that I've seen before include "Run and see" , "Count" , "Depend (upon)" , "Have confidence or faith in" , "Old rally" .)