Overdue improvement on the side (9)
I believe the answer is:
laterally
'on the side' is the definition.
(laterally means on the side)
'overdue improvement' is the wordplay.
'overdue' becomes 'late' (similar in meaning).
'improvement' becomes 'rally' (recovery after illness or decline).
'late'+'rally'='LATERALLY'
(Other definitions for laterally that I've seen before include "Sideways" , "Approaching the side" .)