Wander through Italian city reportedly (4)
I believe the answer is:
roam
'wander' is the definition.
(I know that wander can be written as roam)
'italian city reportedly' is the wordplay.
'italian city' becomes 'Rome' (city in Italy).
'reportedly' shows a homophone (sound like).
'rome' is a homophone of 'ROAM'.
'through' acts as a link.
(Other definitions for roam that I've seen before include "'Wander, ramble (4)'" , "Meander" , "Move about aimlessly" , "Stray" , "Walk without a specific destination" .)