Thank you ,how can I improve on this
"I fully understand your motivation to take this to court however I implore you to reconsider, I have never been charged with any criminal offence before and a criminal conviction would affect my career prospects.
I hope we can come to an amicable agreement and would like to offer to pay the outstanding fare and contribute to your costs in order to settle the matter without incurring additional costs or using your valuable time. I would be willing to pay any costs that has incurred in investigating this matter so far, as well as the fare due. My offer is of £200 towards this or any figure that is suitable to you. As well as the settlement, I also promise that I will always ensure I have ticket on every journey
I would again like to express my deep regret regarding this incident and feel ashamed I caused this unfortunate situation to come about, a huge lesson has been learned from this episode"
My main comment would be that even if you don't want to challenge them on the byelaw, you might still make use of or adapt the first two paragraphs of
@ForTheLoveOf's draft - meaning you could include the circumstances, as well as perhaps a screenshot of the text.
However, I'm still not clear whether telling them you intended to pay on arrival might make them look on your case less favourably if you didn't subsequently do that. Perhaps they don't care as they were going to be in touch anyway.
It might be best to leave out the sentence "My offer is..". It's covered in the previous sentence, and the way it works means that offering a fixed amount such as £200 isn't really relevant - they're supposed to tell you what their costs amount to. If they would otherwise offer a settlement of far less, they might possibly wonder whether you are offering more because you are guilty of more. So you can leave the amount open.
As well as saying you've never been charged, you could say you've never done anything like this before if that's true.
If you say a few words about why a conviction would be a particular problem for you in your career, that might be useful - but best to check that what you say is accurate as it's easy to make assumptions about that.