Observing nought after losing heart (6)
I believe the answer is:
noting
'observing' is the definition.
(note can mean to observe or notice)
'nought after losing heart' is the wordplay.
'nought' becomes 'nothing' (synonyms).
'after losing heart' means to remove the middle letters.
'nothing' with its middle removed is 'NOTING'.
(Other definitions for noting that I've seen before include "Making brief record as memory aid" , "Observing and jotting down" , "Observing or taking down" , "Paying close attention to something" , "Seeing" .)