Leftovers have Irish women in tears (8)
I believe the answer is:
remnants
'leftovers' is the definition.
('remnant' can be a synonym of 'leftover')
'irish women in tears' is the wordplay.
'irish women' becomes 'mna' (I can't explain this - if you can you should give a lot more credence to this answer).
'in' means one lot of letters goes inside another.
'tears' becomes 'rents' ('rent' can be a synonym of 'tear').
'mna' put inside 'rents' is 'REMNANTS'.
'have' acts as a link.
Can you help me to learn more?
(Other definitions for remnants that I've seen before include "Pieces of fabric left over" , "Surviving fragments" , "Pieces at the end of rolls of fabric" , "Stern man has left-over pieces" , "Remains" .)