Most states offer an online claim form for damage done to your vehicle for things like potholes. That said, you won’t be able to ask both the state and your car insurance company to pay for the damages. You’ll need to decide what the best option is for you based on the cost of repairs.
If you decide to submit a claim to your insurance, your rate could increase. Should that occur, consider shopping around. Every insurance company rates policies differently, so you may be able to find a lower rate with another provider.