The lion's running after a girl (6)
I believe the answer is:
alison
'girl' is the definition.
'the lion's running after a' is the wordplay.
'running' is an anagram indicator.
'after' is a charade indicator (letters next to each other).
'lions' anagrammed gives 'lison'.
'lison' after 'a' is 'ALISON'.
(Other definitions for alison that I've seen before include "lady" , "Girl's name" , "female" , "woman" , "Girl" .)