… as well as annoying both Italy and Argentina (4,3,7)
I believe the answer is:
into the bargain
'as well' is the definition.
(I've seen this before)
'annoying both italy and argentina' is the wordplay.
'annoying' indicates an anagram.
'italy' becomes 'i'.
'and' means one lot of letters go next to another (I've seen this in other clues).
'both'+'i'='bothi'
'bothi' after 'argentina' is 'argentinabothi'.
'argentinabothi' with letters rearranged gives 'INTO THE BARGAIN'.
'as' acts as a link.
(Other definitions for into the bargain that I've seen before include "as well" , "Also" .)