For those concerned, geiko and maiko are entertainers, that's all, not even paid escorts. They dance, sing, play music, they make conversation with the patrons of the tea house, and play games - nothing more. Geiko and maiko from Kyoto (the manga's setting) do not engage in sex work. Also, mizuage does not exist in the modern day, so there's no danger for Momohana of that either.
The Westerner's misconception that geiko perform sex work comes from regular prostitutes marketing themselves as geiko towards Allied occupation soldiers. Plus, geiko have historically been associated with oiran (courtesans, probably the actual closest thing to a paid escort at the time, but even then, they're like... extra. They can be differentiated by their more elaborate kimono, hairstyle, and the knot of their obi sitting at the front).
Geiko and maiko play the music while oiran conduct their, uh...business. Nowadays, oiran don't even conduct business (the most I've seen/heard of them are in processions/historical recreations) so now it's just all music and no funtimes.
IIRC in the modern day there are other geiko districts that allow geiko to sleep with patrons that they like, but I highly doubt that any of the districts in Kyoto (very old, very prestigious, very traditional) would allow their own geiko to do that.