End of The Times. End and destruction (13)
I believe the answer is:
extermination
'destruction' is the definition.
(I know that extermination is a type of annihilation)
'end of the times end' is the wordplay.
'end of the times' becomes 'ex' (I can't explain this - if you can you should give a lot more credence to this answer).
'end' becomes 'termination' (termination is a kind of end).
'ex'+'termination'='EXTERMINATION'
'and' acts as a link.
(Other definitions for extermination that I've seen before include "Abolishment" , "Execution" , "Annihilation" , "Process of killing vermin, say" , "Favourite Dalek activity" .)