The Fixed Penalty Notice scheme causes confusion for many people who wish to accept the Fixed Penalty Notice but also wish to plead mitigation (i.e. wanting to present their particular circumstances to the Court to have a lesser penalty) at the same time.
A Fixed Penalty Notice is actually a "conditional offer" from the prosecution. In the majority of cases the proposal is 3 penalty points and £60 fine. There is no flexibility on this and thus a plea of mitigation is not relevant. You either take the proposal or if you wish to argue about the level of fine/penalty points, you reject the offer and appear in Court instead. A summons will be issued to allow you to do this.
You are offered the minimum number of penalty points and a "nominal" fine. This type of fine does not include means testing. If you know you have committed the offence 3 points and a fine of £60 is most likely the best result that you are going to get. To take the matter further can often result in higher punishment and substantially more expense - particularly if the Court imposes its right to means test you before deciding on a fine. Of course if your licence is at risk from totting up too many points or you did not commit the offence you must appear in Court to contest the matter.
To contest Notices issued by the Police the usual method is for the Fixed Penalty option to be withdrawn whereupon you will be summoned to the Magistrates' Court where the case will be heard.