Erase old joke finally giving offence (7)
I believe the answer is:
expunge
'erase' is the definition.
(to expunge is to erase or remove)
'old joke finally giving offence' is the wordplay.
'old' becomes 'ex' (prefix meaning former or onetime).
'joke' becomes 'pun' (pun is a type of joke).
'finally' says to take the final letters.
The final letters of 'giving offence' are 'ge'.
'ex'+'pun'+'ge'='EXPUNGE'
(Other definitions for expunge that I've seen before include "Wipe out" , "Completely obliterate" , "Annihilate" , "Obliterate - erase" , "Delete from the record" .)