Did you receive a fixed penalty notice or were you convicted by a court?
If you only received a fixed penalty notice and paid it within the time requested, then you did not even receive a conviction and are extremely unlikely to have this held against you in any way, provided that you don't have any other recent fixed penalty notices.
If you were convicted by a court, then you will need to declare the conviction on your application. The Home Office says that it will normally disregard a single unspent conviction for a minor offence of a regulatory nature that results in a small penalty.
The offence that you describe might regarded as a minor offence of a regulatory nature. You must, in any case, declare such a conviction on the application form.
The problem with your offence is that the name of the offence is "Driving otherwise than in accordance with a licence". The offence might seem to be of greater severity because it could include contraventions that are more severe than mere failure to put an L plate on your car on a single occasion. If you did receive a conviction and do apply, then you should definitely explain the nature of the offence so that it does not seem more serious than it was. You should also note the amount of the fine.
You should take a look at the UK Border Agency's page on the good character requirement