Welcome to the forum!
Transport for London are less forgiving when it comes to this sort of thing than their National Rail counterparts. I would send a short, concise response mentioning the following points:
- That you are sorry for what has happened
- What you have learned from the incident
- That you are keen to settle the matter without the need for court action
- Offer to pay the outstanding fare and the train company's administrative costs in dealing with the matter
I would not mention that you are a teacher as they could take the view that you should have known better.
Do you mean you'd paid for your journey when you tailgated through the barrier or that you'd paid for previous journeys you made before you were caught.
If you did have a valid ticket then you should tell TfL this in your reply (and enclose evidence of it eg a journey history). Showing you'd paid for journeys you took in the past is unlikely to help you, they want to know about the journey on the day you were caught.
What exactly do you mean by this? A Travelcard is a ticket that's valid for unlimted travel across the fare zones it been purchased for. Was the Travelcard valid for the journey you made - again if it was then tell TfL this is your reply and give details of the Travelcard. Or do you mean you possess a railcard which offers a discount on certain rail tickets (for example a 16-25 Railcard)?
Transport for London don't usually offer out of court settlements they tend to prosecute under the TfL Bylaws but might offer a final warning for more 'lower level' cases.
Here's a link to TfL's Revenue Enforcement and Prosecutions Policy which is worth reading before you send your reply.
Feel free to post a copy of your draft reply in this thread and forum members will be happy to proof read it for you.