Puts up with a sailor on a date in Rome (6)
I believe the answer is:
abides
'puts up with' is the definition.
('abide' can be a synonym of 'put up with')
'a sailor on a date in rome' is the wordplay.
'a sailor' becomes 'ab' (abbreviation for able seaman).
'on' is a charade indicator (letters next to each other) (in a down clue, letters appear on others).
'a date in rome' becomes 'ides' (Roman calendar term, as in the Ides of March).
'ab'+'ides'='ABIDES'
(Other definitions for abides that I've seen before include "bears" , "Dwells (archaic)" , "Lives or tolerates" , "Stomachs" , "Continues without fading or being lost, memory perhaps" .)