I need to send the letter off tomorrow, this is the closest I've got:
Dear Sir/Madam,
I am writing in response to receiving a Notice of Intention to Prosecute Letter on 12/09/2019, Reference: XXXX.
Please note that my name is <NAME>, not <INCORRECT> as your correspondence reads.
On 26/05/2019, I travelled from New Southgate (NSG) to Hatfield on the 10:52 service in order to attend “Slam Dunk Festival'. On reaching the platform at New Southgate, I used Contactless to start the journey. Upon reaching Hatfield, the station was very busy and I attempted to ’tap out' at the Hatfield barriers. The barriers did not open and I was informed that I must pay £20 to exit through the gate. I mistakenly believed that Hatfield accepted Contactless and mistakenly believed the Penalty Fare was issued unfairly. I thought I had already paid via the Contactless machines (I was charged the full day fare from Transport for London on this date as my TfL journey from NSG was not terminated).
This was a clear error of judgement on my behalf and refusing to pay the Penalty Fare was a regrettable course of events that I am extremely remorseful for. I am an infrequent traveller to London, had never travelled on that line or between those stations before and believed that due to it's proximity to London, I would be able to use my Contactless card to complete my journey when I arrived in Hatfield. I regrettably refused to pay the Penalty Fare, as I genuinely believed I had paid for my journey, for which I send my sincerest apologies.
I would like to apologise unreservedly, and settle this matter if I may. I am prepared to offer you the full single fare for my journey from New Southgate to Hatfield, the Penalty Fare, plus any reasonable costs incurred in pursuing your claim against me.
Since this incident, I have moved address. My mail is being forwarded, but please send further correspondence to:
<NEW ADDRESS>
Yours faithfully,
<NAME>