To be pedantic, it's not a 'fine' but a 'fixed penalty'. It doesn't form part of a criminal record and, as such, there's no rehabilitation period. i.e. if a potential employer, or an insurer, asks if you've ever been convicted of a criminal offence you can lawfully answer 'No'. (The only exception would be in certain occupations, such as working with children or vulnerable adults, where a DBS check is required).
The piece of paper you were given is similar to the form which would be sent to you after a minor speeding offence. It invites you to accept that you committed the offence, and to pay the fixed penalty. However you don't have to do so; you are free to 'have your day in court' and to challenge the evidence. But once you've paid the fixed penalty you've admitted the offence and you can no longer challenge it.
Your chances of getting the penalty revoked, or reduced, are effectively zero. (After all, it's only the same penalty as for dropping a fag end on the pavement). Any legal challenge would almost certainly cost you thousands of pounds and be completely pointless anyway.
Chris