tend?
(Heading off offer = tend(er))
change = tend (the wind is tending to the south).
I don't know where Rome's third Prime Minister might fit, though.
I think i was wrong. Sorry.
i thought rome's 3rd -m
heading off offer- (t) end
change- mend.
however, seem to have missed the prime minister reference completely :-\
I think that clue has stuffed a lot of people today.
The chances are, especially with the Times Crossword, that if you can't explain every word of the clue then you may well be wrong.