Pole will bore Italian city then go to bed (4,2)
I believe the answer is:
turn in
'go to bed' is the definition.
(synonyms)
'pole will bore italian city' is the wordplay.
'pole' becomes 'n' (abbreviation for North, as in North Pole).
'will bore' indicates putting letters inside (to bore something can mean to drill into it).
'italian city' becomes 'turin' (Turin is a city in Italy).
'n' put inside 'turin' is 'TURN-IN'.
'then' is the link.
(Other definitions for turn in that I've seen before include "Hit the hay" , "rat on" , "Informally, go to bed or hand over" , "Retire to bed" , "Go to bed (informal)" .)