Close behind them, improving (8)
I believe the answer is:
emending
'improving' is the definition.
(emending is a kind of improving)
'close behind them' is the wordplay.
'close' becomes 'ending' (synonyms).
'behind' is a charade indicator (letters next to each other).
'them' becomes 'em' (abbreviation).
'ending' put after 'em' is 'EMENDING'.
(Other definitions for emending that I've seen before include "Removing errors and making improvements" , "... putting things right" , "Setting right, correcting" .)