I think "a su hora" means more like "at the correct time for his meal", though that's definitely not the best way to phrase it! (Maybe "at the appointed time"?) Then, "en un tiempo determinado" would be "within a set/established time(frame)/schedule".
So I'd put: "The child should eat well, eat a balanced selection, at the correct time and within a certain timeframe."
Hope this helps at least understand the idea, though probably not the proper way to express it all!