Greek character, Hero's lover, missing sweetheart's flirt (9)
I believe the answer is:
philander
'flirt' is the definition.
(synonyms)
'greek character hero's lover missing sweetheart's' is the wordplay.
'greek character' becomes 'phi' (letter from the Greek alphabet).
'hero's lover' becomes 'leander' (I've seen this in another clue).
'missing' suggests deleting specific letters.
'sweetheart' becomes 'e' (the heart/centre of 'sweet').
'leander' with 'e' removed is 'lander'.
'phi'+'lander'='PHILANDER'
(Other definitions for philander that I've seen before include "Womanise" , "Flirt, chase women" , "Have an affair" , "Play the field" , "Womanize" .)