Damp has to destroy finally (5)
I believe the answer is:
musty
'damp' is the definition.
(thesaurus)
'has to destroy finally' is the wordplay.
'has to' becomes 'must' (I've seen this before).
'finally' indicates one should take the final letters.
The last letter of 'destroy' is 'y'.
'must'+'y'='MUSTY'
(Other definitions for musty that I've seen before include "Stale, mouldy-smelling" , "Frowsty" , "Mouldy; outdated" , "Smelling stale or damp" , "Smelling of mould" .)