Head started to lose heart (4)
I believe the answer is:
bean
'head' is the definition.
(I know that bean is a type of head)
'started to lose heart' is the wordplay.
'started' becomes 'began' ('begin' can be a synonym of 'start').
'to lose heart' means to remove the middle letters (I've seen 'lost heart' mean this).
'began' with its middle taken out is 'BEAN'.
(Other definitions for bean that I've seen before include "Vegetable; coffee seed" , "Edible seed" , "coffee, perhaps" , "popular character with kids" , "Edible legume" .)