Thanks left half for a story (4)
I believe the answer is:
tale
'for a story' is the definition.
('tale' can be a synonym of 'story')
'thanks left half' is the wordplay.
'thanks' becomes 'ta' (UK informal version of 'thank you').
'half' means to take half the letters.
'left' cut in half is 'le'.
'ta'+'le'='TALE'
(Other definitions for tale that I've seen before include "Fable" , "Anecdotal narrative" , "Narrative - fabrication" , "See 20" , "See 12" .)