On the radio, closely follow story (4)
I believe the answer is:
tale
'story' is the definition.
(I know that story can be written as tale)
'on the radio closely follow' is the wordplay.
'on the radio' shows a homophone (sound like).
'closely follow' becomes 'tail' (as in tailing somebody).
'tail' is a homophone of 'TALE'.
(Other definitions for tale that I've seen before include "Report" , "'Story, narrative (4)'" , "Anecdotal narrative" , "Fable" , "A story, perhaps out of school" .)