Started to lose heart -- and head! (4)
I believe the answer is:
bean
'head' is the definition.
(bean is a kind of head)
'started to lose heart' is the wordplay.
'started' becomes 'began' ('begin' can be a synonym of 'start').
'to lose heart' suggests removing the centre (I've seen 'losing heart' mean this).
'began' with its middle taken out is 'BEAN'.
'and' is the link.
(Other definitions for bean that I've seen before include "coffee, perhaps" , "Pulse" , "Seed; head" , "It's often baked" , "Haricot, for example" .)