Are there any natural cures for hypothyroidism in a child?

Yes. Although there are "natural" treatments for hypothyroidism, the standard of care is a synthetic (man-made) hormone that mimics the hormone the body is not making correctly. Low thyroid levels in children can permanently damage brain development, so it is important to take medicine that is well proven to work. Don't risk taking something "natural" that might not work. Natural isn't always better!
No. The natural cure is thyroid hormone which is really natural same as the body makes.