If the pothole is on a city street, the city may be responsible for your car damage, and you may be able to get reimbursed for the damage by the city. However, the reimbursement process can be lengthy, so if your vehicle isn't drivable, it may make more sense to file an insurance claim for the pothole damage or pay for the repairs out of pocket and then file for reimbursement from the city later.
County and state governments may also reimburse you for pothole damage if the pothole is on a road they manage. Bear in mind, the time and effort needed to get reimbursed generally makes it not worth pursuing for minor damage. Plus, not all jurisdictions provide compensation for drivers whose cars are damaged by potholes.