Washed-out end to our month in Italian city (7)
I believe the answer is:
palermo
'italian city' is the definition.
(city in Italy)
'washed-out end to our month' is the wordplay.
'washed out' becomes 'pale' (washed out can mean pale or faded).
'end to' indicates one should take the final letters.
'month' becomes 'mo' (abbreviation for month).
The final letter of 'our' is 'r'.
'pale'+'r'+'mo'='PALERMO'
'in' is the link.
(Other definitions for palermo that I've seen before include "Sicilian capital" , "Sicilian port" , "Sicilian city" , "Chief city of Sicily" , "Italian city" .)