Girl’s run off after writer (8)
I believe the answer is:
penelope
'girl's' is the definition.
'run off after writer' is the wordplay.
'run off' becomes 'elope' (synonyms).
'after' is a charade indicator (letters next to each other).
'writer' becomes 'pen' (something which writes).
'elope' after 'pen' is 'PENELOPE'.
(Other definitions for penelope that I've seen before include "Girl's name" , "Waiter" , "Odysseus's wife" , "Wife of Odysseus or Ulysses" , "Woman" .)