Possibly late with a story (4)
I believe the answer is:
tale
'a story' is the definition.
('tale' can be a synonym of 'story')
'possibly late' is the wordplay.
'possibly' indicates an anagram.
'late' anagrammed gives 'TALE'.
'with' is the link.
(Other definitions for tale that I've seen before include "Narrative - fabrication" , "Story; lie" , "Yarn or story" , "See 12" , "See 20" .)