How can I fry leftover mashed potato?

For one half of your problem: You can add a little bit of milk to dry mashed potato to make it moister, if it's too dry to use in a given application. The other half is a little more difficult, but instant mashed potato can be used to sort of "soak up" some of the moisture to make it drier.

When life throws lemons at you, make a lemonade!!

I know this saying is intended in a different way, but what I am trying to start with here is, when you have got mashed potatoes on hand, go try all these yummy recipes which I am going to suggest here now.

To start with, I am assuming that the mashed potatoes that we are talking here is made up of just potatoes(of course), a dab of butter and minimum dressing say salt and pepper and the texture is still gooey and not on the wetter side! With this on hand, you can try out these following recipes considering you want to more than fry your mashed potatoes:

1. Leftover Mashed Potato Sandwich: Take a pan and drop little oil in it(2-3 drops). When the oil heats up add some tempering like cumin, chilli flakes, a spoonful cut onions, minced garlic, green chillies etc (basically for flavour). Then throw in the mashed potatoes and saute everything together until the potatoes start becoming very light golden brown. You might need to stir it so that it doesn't stick to the pan a lot. You can use this mashed filling inside any veg sandwich or sub. Just put a layer between bread slices, add your favorite veggies and cheese and you have an awesome sandwich in front of you.
2. Indian Potato Curry: If you are a fan of indian food and you have got any indian spices like cumin, asafoetida, turmeric powder, chilli powder, coriander powder in your closet, then you could try making this quick curry. Start with oil in the pan thing and let it heat. Add cumins and let then crackle. Then add a am of all these powder masalas directly in the oil and then put the potatoes, 2-3 spoons yoghurt and water and let everything come to a boil. This can be eaten with hot rice or rotis!
3. Puff Potato Pasties: If you buy puff pastry sheets and know how to use it, you could use the potato sandwich mix and roll it into a puff pasty sheet and put it in the oven to make a yummy snack item!
4. Potato Patty: You can add some corn flour or refined wheat flour in the mashed potatoes along with some spices like chillies, garlics, and other flavoring agents and make it into a dry dough. Then you can make balls out of this dough and flatten it to deep fry until dark golden brown. These patties can be used in a burger instead of meat patties.

Maybe I gave you enough ideas to try next time!

For instance could I later add cheese or even an egg or some other bonding agent so it stays together in the pan long enough to seal and survive being manipulated a little with an egg-flip?

Yes, use an egg. It will dramatically affect how well things hold together.

Depending on the intended application, flour might also be suitable to help dry it out. (eg, gnocchi, boxty, matafan)

As for cheeses, they can either add moisture (eg, ricotta) or help dry it out (parmesan or other finely grated hard cheese)