Let the baby bird eat all it wants. Keep that cup full, maybe get a larger cup. Keep food (and water) available at all times.
I have a baby pigeon that's about 3 weeks old and he constantly wants to eat. I feed him grains (wheat) and he seems to like it but 1-4 hours after feeding him as much he will consume, he gets up and starts making a squeak noise and starts pecking me and eating things off the floor until I feed him again. I usually feed him 4 times a day, as much as he will consume each time per feeding. Is that too much? Why is he always hungry?