You should reply with all the evidence you can find to demonstrate that you are a regular fare-paying passenger. There is a substantial difference between the Offence which the Company will persue against you for (simply) not having a valid ticket when requested, and on the other hand, evading a fare (which can be demonstrated by not buying one when there was an opportunity - whether or not that was due to having no money and/or no Oyster credit).
If it realy is a first time Offence, and you can demonstrate this, then a statement of facts and an apology will set the right context for the best outcome for you. You should also try hard to recall what you said to the Inspector whenthey 'took some details', That will now be a Witness Statement on file and could be used to incriminate you. It will not help if anything you write in your reply is inconsistent with that Statement.
For a similar reason, I strongly advise you
not to take the following advice (which I will guess comes from someone with no experience in Criminal case management):-
Speak to customer services AND get legal advice (can get this from either the CAB or if you happen to be a member of the Cooperative Group, they offer free legal advice) in case they still pursue legal action.
It is NEVER advisable to take professional advice
and concurrently attempt to negotiate personally with the Prosecution.
Never.
It will, perhaps, be helpful to seek advice from a local Law Firm specialising in general Criminal Defence work; but a specialist Law Firm with Railway Law expertise will probably cost you £3000 - £5000. Clearly that is quite disproprtionate to the potential penalties.
There are advisors here who can help to point you in the right direction and perhaps minimise your exposure to risk (such as not speaking to the Prosecution before instructing a Solicitor!). I expect we can assist in avoiding Prosecution of the more serious claim of intention to avoid payment if you are, evidentially, a very regular paying passenger.
