Quizlet seems like a perfect fit for custom list learning.
This flashcard app allows you to create lists (or chose from millions of existing ones that other users created) that can contains questions, words, diagrams and much more. Progress can be tracked with several statistics in a modern and intuitive UI. Under the hood, proven learning methods and a bit of AI is used to chose the questions that will maximize the user learning. Quizlet provides out-of-the-box support for a students-teacher setup, which might be suited for a parent-child setup too.
There are a few drawbacks, though. For teachers, the free period of the freemium is quite reduced. Correct and proposed answer must match completely, which can be quite anoying. Keep in mind that Quizlet is not a language learning app by itself, but rather a very friendly yet powerful tool to memorize a lot of things.