“It was an oversight” seems a bit vague....
If you admitted more when interviewed it might be seen as you now trying to minimise the offence and not fully accepting responsibility.
So, it also depends on
a) the circumstances (what really happened?), and
b) what you said had happened at interview .....
How about this as a response:
Dear Fare Evasion Manager,
With regard to your letter dated 18 November, I would like to sincerely apologies for purchasing the incorrect ticket for my journey.
This was an oversight on my part and one which I am very embarrassed about.
I can assure you this will not happen again and I would like to offer to pay the difference in fair as well as the administrative costs incurred by GTR.
I apologies again for the inconvenience caused.
Should I go in to more detail if this is tok short and missing the point ?
Don't touch them owe me £500 since January 2019 make excuse after excuse. Seem they always have software problems sending money out. Keep saying they will call back or email nothing been chasing it now for 6 mths the phone staff always have the same banter we will chase it up and get back to you then nothing!