The circumstances you describe seem to fit exactly into the format of a little deception that is practiced by thousands of rail passengers every day - i.e. travelling without a ticket and when asked, then giving the name of a station at which they boarded which is closer to their destination. It is estimated to cost the industry more than a hundred thousand pounds every day.
It is also captured by a criminal offence under Section 5 of the Regulation of Railway Act "Penalty for avoiding payment of fare". Some Railway Companies bring forward prosecutions against passengers more vigourously than others. Those that operate large numbers of short distance commuter services (which includes Northern Rail) put regular investigations in place to detect this deceipt.
The consequence is usually that an Investigating Officer makes further enquiries (including the response to the letter you have received) and on receipt of all available evidence then makes a two part test: a) is there adequate evidence to give a high probability of success in a prosecution; b) is a prosecution in the public interest.
From your report on here, the answer to both of these will be in the affirmative. A prosecution under S.5 of the RoRA is a 'recordable offence' in the Police National Computer.
But that is no reason not to reply with a suitably apologetic response, recognising the gravity of the situation, what you have learned from this incident, how your behaviour and attitude has changed as a result, and saying nothing about 'not breaking a 20'.
It is possible that if you have no prior record of having attempted a fraud such as this, then the Investigating Officer might be inclined to adopt a less punitive outcome. Otherside the typical level of fine from the Courts for a deception such as this is around £150 - £200 depnding on means. Plus costs of perhaps £120 and Victm Surcharge.
So far, so bad.
The good news is that Northern Rail have put in place a scheme refered to as 'Failure to Pay' or FTP. Some people on this forum are very critical of this scheme and refer to it as "Penalty Fakes". I am not one of them. This scheme identifies first time offenders and introduces a structured alternative to a Prosecution, and passes the choice to the passenger. On the basis of what you have written on here, I think there is a resonable prospect that this offer might be given to you.
That would be the offer to pay an administrative settlement of £80 plus the unpaid fare to discharge the liability to answer to the Courts for the Crime. If they do make this offer, then you are under no obligation to accept it and could attempt some sort of defence if you wished, but from what you have said on here, I cannot advise you to seek that option. If you are offered an administrative settlement, I would urge you to accept it.
Finally, I'm sorry that some people on this forum take the opportunity to criticise others who offer sound advice. I hope you are able to learn what you need to know from what has been posted in response to your request.