If you have a heavier flow, you will have to dump the cup mid-day at least once. I dump, wipe it inside and out with a bit of toilet paper, and shove it back in. It's fine to wait to wash it properly until you are back home.
Sometimes, at work, I used to dump it, wipe it, then covertly hold it in my hand to wash it in one of the sinks in the row, then gone back in a stall to re-insert. I would only do it when I didn't think there was anyone else in the bathroom, and it didn't seem to be much improvement over just wiping out the cup.
But, in retrospect I'm not sure that it would have been a big deal if anyone else saw the cup, as long as I didn't bring a huge amount of blood to the sink with it. It was a women-only bathroom, and it's not like the sinks are used for anything but post-toilet washing.