"a child is diagnosed with an enzyme deficiency that prevents the production of hydrogen peroxide. what would the likely outcome be of such a deficiency? "

Peroxisomes depend on hydrogen peroxide in order to function so an enzyme deficiency that ends in the inability to form hydrogen peroxide would likely end up unable to digest things that have long chain fatty acids. These acids would instead build up in peroxissomes until the majority of the cellular contents were displaced by oversized peroxisomes. This would finally end up in cell death.