The second paragraph reads as if it is AI generated verbiage, and could all go, except for possibly the last sentence.
The penultimate paragraph doesn’t feel right either. Carrying a forged railcard for months doesn’t feel like an isolated lapse. And I’m not sure what you mean by a fair and respectful manner. Are you asking them to be fair and respectful. Fairness doesn’t really come into it, they are entitled to prosecute you, and I’m not sure how you can promise to be fair?
My view may be overly harsh but for what it is worth:
1. Paragraph 1 The use of the word incident (singular). I think it would be better expressed as ' for my use of an invalid railcard'. This wording does not state that it was a single incident though it could be taken to mean that. You could of course insert the word "repeated" so it reads "repeated use of an invalid railcard.
2. Paragraph 2. Using an invalid railcard and being caught is not unfortunate.
3. Paragraph 2. Not sure why purchasing an ouster card will give them any comfort that you wont reoffend.
4. As
@Brissle Girl has said it doesnt feel like an isolated lapse. In fact it most definitely wasnt. It was a conscious decision to avoid paying the correct fare by altering a railcard. This demonstrates prior thought and action.
5. Paragraph 4. I think you need to say that you are unable to fully access your booking and travel history. Your railcard expired on (insert date) and that since then according to your bank statement you spent £xxx. Say that you would be grateful if they could provide a list of tickets purchased since (insert date) and that you will endeavour to reconcile them to booking confirmation emails.
6. Paragraph 5 repeats part of paragraph 3. The bit about a fair and respectful manner is nonsense.
7. The last sentence can be abbreviated to I look forward to hearing from you.
8. Thank them for their understanding and assistance in resolving this matter. (You dont want to suggest that it is them cooperating with you).