Welcome to the forum!
Thameslink are one of the more pragmatic companies at dealing with this sort of thing. I expect you've been sent their standard initial letter but can you upload a copy of it in this thread (with personal details redacted) so we can be certain.
If you have been sent their initial letter then I suggest you reply with a short, concise letter 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
Govia Thameslink Railway normally offer an out of court settlement to people who engage with the process, and who haven't come to their attention before. This generally costs around £150 plus the outstanding fare. There's no guarantee of this and they could decide to send the case to the Magistrates Court if they want to. An out of court settlement might appear to be a fine, but it isn't (only a court can impose a fine as a punishment following a conviction) and you won't have a criminal record as a result of accepting one.
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.