I don't think that including it would add anything to your story. I agree you should keep it simple.
cheers, this is the draft I have with improvements, I've also added a bit on the end about my change in address. Another point, is it worth me mentioning the fact I've had a previous penalty fare on this route, or is it best I just say nothing and hope they overlook this.
Dear Sir/ Madam,
I am writing in response to your letter on the … June 2024, thank you for offering me the chance to respond.
On the … of June 2024 I travelled from Burley Park to Leeds Station. I arrived at the station and attempted to buy a ticket on my phone for the train at the platform but was unable to do so since it was a delayed service. As a result, I boarded the train and bought a ticket for the next train thinking it would suffice. I would like to apologise for not having a correct ticket for my journey before boarding the train.
I am now aware a ticket must be bought before boarding the train and take full responsibility for this error. I have learnt an important lesson and will ensure I am better prepared and more organised when purchasing tickets in the future. I am eager to settle this matter without court action and would very much appreciate the opportunity to pay the outstanding fare and any administrative costs incurred by Northern Trains for dealing with this matter.
I would also like to kindly ask if further correspondence could be please sent to my home address which is …, as I am in the process of moving out of my student house, which is the current address you have.
Thank you for your understanding consideration, I look forward to you hearing from you.
Kind regards,