Is your child covered on your auto insurance if you give them permission to drive your car?

If you have them listed as a driver on your policy then yes they will be covered. If you do not have them listed on the policy then you have violated a primary term of the policy and committed material misrepresentation. When this happens the insurance company has the right and often does deny the claim. You have not abided by the contract terms where you agreed to list all household members above 15 years old and have not paid any premium to cover the child driving a vehicle.