Gradually destroy (7)
I believe the answer is:
corrode
'gradually destroy' is the definition.
(to corrode is to destroy over time)
This is all the clue.
(Other definitions for corrode that I've seen before include "Gradually weaken (metal)" , "Gradually weaken (by chemical action)" , "Growth in hazel buds" , "Slowly eat" , "Cause to rust" .)