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Turn based rpgs mostly although you can of course connect a Bluetooth controller and the psp has really nice library .
Jeanne D’Arc, Disgaea 1 and 2, final Fantasy Tactics, Tactics Ogre would strike me as the most touchscreen friendly type of games being trpgs.
Not sure how much I can actually say on that and I know Nintendo been running rampant on things like that but I’d say by far the best emulator is PPSSPP and you either need iso files or psp I think exclusively cso files (compressed isos).
I suspect chd might work as well but mine are all cso.