Bad-mouth picture put up by Mussolini (7)
I believe the answer is:
traduce
'bad-mouth' is the definition.
'picture put up by mussolini' is the wordplay.
'picture put' becomes 'art' (I can't justify this - if you can you should believe this answer much more).
'up' says the letters should be written in reverse (in down clue: letters go upwards).
'by' says to put letters next to each other.
'mussolini' becomes 'duce' (I've seen this before).
'art' reversed gives 'tra'.
'tra'+'duce'='TRADUCE'
(Other definitions for traduce that I've seen before include "'Malign, badmouth (7)'" , "Slander - misrepresent" , "Tell lies about (someone)" , "Speak badly of" , "Speak ill of" .)