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If you can take a picture of the pothole, damage to your car/bike or you as well as any repair charges. Also if possible see if it has affected anyone else and make a joint claim as this is a lot more powerful.
Report the pot hole to the council in that area, they control 90% of the roads. You can do this directly, or use resolver.co.uk to help you manage your case. If the road is not maintained by the council they will tell you who you need to contact, such as the Highways Agency (motorways and A roads). Remember to always be polite and professional as well as reasonable throughout the process.
The council should give you feedback on if they will pay you compensation. If they do not give you money towards the damage you have caused, then you make a Freedom of Information Request about what they have paid out for when consumers have made claims for potholes. You can see then what they have paid out for.
Ask for their decision to be appealed. If they refuse request mediation of the issue with the council.
If you get nowhere
You can consider legal action, but get advice first and keep a complete record of your case throughout the process. You can always claim on your insurance if you have it, but of course this may affect your premiums and any no claims bonus.