How do you dispose of broken mayim achronim cups? Mine was plastic and it fell and broke and cannot be fixed. Thanks!
Items once used for a mitzvah (tashmishei mitzvah) that wore out do not require genizah, according to the letter of the law. Mishnah Berurah 21:1 (א) חוטי ציצית וכו' - וה"ה לכל תשמישי מצוה כגון סוכה ולולב ושופר וכל כה"ג לאחר שנתבטלו ואינם עומדין עוד למצותן:
But it is still good to put them in genizah, and not just chuck them straight in the garbage, according to the Rema ad loc. This is a way of showing respect.
There is a Halachic disagreement whether mayim achronim is considered a mitzvah nowadays, so that is an extra reason to be lenient, even for Ashkenazim.
A practical solution would be to wrap it up carefully before placing in the garbage container, assuming it is not practical to place it with the Sheimos.