Gradually destroy a stick in the middle (7)
I believe the answer is:
corrode
'gradually destroy' is the definition.
(to corrode is to destroy over time)
'a stick in the middle' is the wordplay.
'a stick' becomes 'rod' (I've seen this before).
'in' means one lot of letters goes inside another.
'the middle' becomes 'core' (core can mean the centre of something).
'rod' inserted into 'core' is 'CORRODE'.
(Other definitions for corrode that I've seen before include "Gradually weaken (metal)" , "Oxidise" , "Chemically weaken" , "Destroy metal gradually by chemical action" , "Cause to rust" .)