Both are accepted. Картопля is a collective noun which can mean one potato or potatoes.
Potatoes is plural of potato, but countable (I suppose). That sentence just sounds right to me to say it in plural.
The thing is that "potato" is used here as a mass noun (uncountable) that describes the kind of stuffing. I believe that "potato" is the mass noun and not "potatoes." When you say "I'm eating dumplings with potatoes," it sounds like you have dumplings and extra potatoes on your plate. However, I'm not a native English speaker, so it would be nice to have a third opinion on this.
I understand that might sound like extra potatoes are alongside the vareneky but it sound ok to me with the plural there. I'm not an English teacher though, I asked someone else and will put their answer here for you when they reply.
I guess your way of saying it is the British way and that is the way many Ukrainian/English teachers say that.
Potato-filled dumplings or potato-stuffed dumplings perhaps? Stuffed dumplings with potato, even.
"Dumplings with potato" sounds odd to me.
Perhaps "dumplings with potato filling" to appease all.
Nope. The application wants you to use the word "with". I tried typing "potato dumplings" to bypass it, as it makes sense, but I still got it wrong.
"Potato dumplings" sounds to me like the dough is made of potato. If I google "potato dumplings" I get recipes of dumplings that are made of potato dough, such as German Knödel or Polish kopytka. Ukrainian varenyky are made of wheat flour, like Polish pierogi, or like Italian ravioli (but larger and shaped differently), or like German Maultaschen (but shaped differently).
A direct object is usually in accusative. "Вареники" is accusative plural in this case.