I've actually made potato patties using leftover mashed potatoes
and no mix ins..not even eggs. Just take cold mashed potatoes and
form them into patties, flour, then keep them cold.

Heat a frying pan with some oil and once hot, add the COLD potato
patties carefully. Let cook and don't touch until they are very
browned and crusty, then carefully turn and brown the other side.
No worries and good potato cakes. No eggs either. :-)