"The mother of the girl does not allow her to run after eating."
Translation:Mẹ cô gái không cho phép cô ấy chạy sau khi ăn.
"Chạy bộ" refers to a more technical aspect and seems to be unnecessarily serious in daily conversations. And yes, you can use either "sau khi ăn" or "sau ăn". I think they're interchangeable. "Sau ăn" would only matter if you want to make your sentence the shortest as it could be.
From the sound of this sentence, we can guess the girl is quite young, like younger than to be called either cô gái or cô ấy. I can't think of a good pronoun to be used here (Bé, maybe? I'm not sure if it's technically a pronoun though. Nó? Well I think it's acceptable.) because I'd use name or repeat the noun in this case.