Said to have begun losing heart (6)
I believe the answer is:
stated
'said' is the definition.
('state' can be a synonym of 'say')
'begun losing heart' is the wordplay.
'begun' becomes 'started' ('start' can be a synonym of 'begin').
'losing heart' says to hollow out the word (remove centre letters).
'started' with its centre taken out is 'STATED'.
'to have' acts as a link.
(Other definitions for stated that I've seen before include "Declared as fact" , "Declared or asserted" , "Expressed in written or spoken words" , "Clearly expressed" , "Specified" .)