Titled man losing heart in Greater London borough (6)
I believe the answer is:
barnet
'greater london borough' is the definition.
(I've seen this in another clue)
'titled man losing heart' is the wordplay.
'titled man' becomes 'baronet' (I've seen this before).
'losing heart' says to hollow out the word (remove centre letters).
'baronet' with its centre removed is 'BARNET'.
'in' acts as a link.
(Other definitions for barnet that I've seen before include "Locks" , "football team" , "N. London borough" , "Hair (rhyming slang)" , "London borough, site of 1471 battle" .)