Write an unemotional but detailed letter to the authority. Remember that your letter might eventually be read out in Court, so think carefully about what you say! Explain exactly what happened, when, where, etc. The purpose of your journey is not relevant, but if you have not used the road before or only use it occasionally, tell them. Include a map (hand drawn is fine) of the area with an 'X' showing where the pothole was.
If your map is inadequate, the authority will normally send you a scale plan for you to mark. Highlight exactly what costs you have incurred and explain that you wish to claim for compensation. If you have gathered your FOI information already, there is no need to tell them about it at this stage.
The first proper response you receive will either acknowledge your letter, inform you that it has been passed to a claims handler, or refuse your claim. At some point you will probably receive a letter refusing your claim, and stating that they have in place a system of inspections, and that under Section 58 of the Highways Act 1980 they have a legal defence to your claim.
Expect this letter, and stay calm when it arrives. Don't telephone the letter writer and/or be abusive - they are people just like you, with families, doing their job in the way they have been told to do it. Authorities are advised by their legal people to send this letter as they know that a very high percentage of people will not pursue their claim any further after they receive it. However, in reality, this is just the start of your claim.
If not done already, now you need your FOI request information, which you will use to assess the authority's defence against your claim.