Beg last of Oxford dons to shut up (5)
I believe the answer is:
cadge
'beg' is the definition.
(I know that cadge is a more specific form of the action beg)
'last of oxford dons to shut up' is the wordplay.
'last of' says to take the final letters.
'dons' means one lot of letters goes inside another (don means to wear or put on).
'to shut up' becomes 'cage' (I've seen this in other clues).
The last letter of 'oxford' is 'd'.
'd' going inside 'cage' is 'CADGE'.
(Other definitions for cadge that I've seen before include "get free" , "freeload" , "Try to get (something) without paying" , "Scrouge, ask for" , "Get by begging" .)