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 anagramming the letters (the letters run around to different places).
'after' says to put letters next to each other.
'lions' anagrammed 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" .)