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