Will a new cat give your child allergies or will it prevent your child from getting allergies; or neither?

Could go either way. Children can often develop allergies to cats without any exposure in the home. They get enough exposure outside of the home to develop an allergy. If your child is already allergic, and you didn't know it, a new cat could cause a reaction. On the other hand, children with high levels of cat exposure may be less likely to develop an allergy to cats.
Neither. The child is either allergic to cat allergens or not. You can have the child tested if you need to know before getting a new cat.