Allergies to cats are believed by many sourced to be cause by saliva, urine and dander. Allergies to pollen brought in on cat's fur have also occaisionally been mistaken for an actual allergy to cat's fur itself. Before I suggest strategies to help combat allergies, I'd like to stress the importance of actually be sure as whether your child has an allergy first and to be sure of what's causing it. If you start tackling the wrong problem you would not only be putting yourself, your children and your cat through unneccesary measures, but your child's condition could be worsening in the mean time.
You can help mittigate the risk of a reaction to urine by making sure your cat is toilet trained and its litter tray is kept away from your children (which you should do anyway). If the cat was able to go outside to defecate and urinate this would be less of an issue as it may be more likely to go further afield to do so. Age plays a factor here as well since older cats are more likely to become incontenant or to be less picky about where they excrete.
A well groomed cat should have less dander and most cats are perfectly capable of grooming themselves unless unhappy or unwell. If you suspect dander to be an issue you can buy cat combs and help groom your cat (but don't force your cat to be groomed if it doesn't like being groomed).
Saliva is harder to guard against, but keeping the child away from the cat's food and away from the cat when it is about to be fed can help. (Animals such as dogs and cats often pick up on signals that give away when they are about to be fed, which tends to cause them to salivate, as proven by Igor Pavlov's experiments in the classical conditioning of dogs). It's harder to keep the child away from a cat when it is grooming, the best way to achieve that would be to teach the child to be careful.
If you suspect your child might have an allergy, you should take the child to a doctor first to confirm the diagnosis. The doctor may also prescribe allergy medication which would be beneficial and perhaps provide additional information.