Destroy pests initially missed in old art gallery (11)
I believe the answer is:
exterminate
'destroy' is the definition.
(exterminating is a kind of destroying)
'pests initially missed in old art gallery' is the wordplay.
'pests' becomes 'vermin' (I've seen this before).
'initially missed' means to remove the first letter.
'in' is an insertion indicator.
'old art' becomes 'ex' (I am not sure about this - if you are sure you should believe this answer much more).
'gallery' becomes 'tate' (Tate Modern, Tate Britain etc).
'vermin' with its initial letter taken off is 'ermin'.
'ex'+'tate'='extate'
'ermin' going inside 'extate' is 'EXTERMINATE'.
Can you help me to learn more?
(Other definitions for exterminate that I've seen before include "Wipe out" , "Destroy utterly" , "Dalek's order to kill" , "Annihilate" , "Finish off" .)