Si pago la multa hoy, puede que me hagan un descuento.

In Spanish, si + present indicative is the normal way to talk about a realistic future condition (something that could happen):

• Si pago... = If I pay... (today / in the future)

Spanish generally does not use the future tense right after si for this meaning. So si pagaré sounds wrong here.

They express different “levels of reality”:

• Si pago la multa hoy, ... → a possible/likely condition (realistic).
• Si pagara/pagase la multa hoy, ... → a more hypothetical/unreal condition (often used like If I were to pay... / If I paid... in a more speculative sense), and it usually pairs with conditional in the other clause (e.g., ... me harían un descuento).

Puede = can / may as a normal verb meaning ability/possibility:

• Puede venir. = He/She can come.

Puede que + verb (subjunctive) is a fixed structure meaning It’s possible that... / It may be that... and it introduces uncertainty:

• Puede que me hagan un descuento. = It’s possible they’ll give me a discount.

Because puede que triggers the subjunctive in the following clause. It expresses uncertainty rather than a sure fact.

• Puede que + subjunctive → uncertainty/possibility
  So: puede que me hagan... (subjunctive of hacer: hagan)

Me hagan literally means “(they) do/make to me”, but in context it means “(they) give me” (a discount). The subject is intentionally vague: they = the authority/company/office handling the fine.

Spanish commonly leaves that “they” implied.

In Spain, hacer un descuento is very common and natural: it means to apply/give a discount.
Dar un descuento also exists and is understandable, but hacer un descuento is often the default phrasing in everyday use (shops, fees, penalties, etc.).

Me is an indirect object pronoun meaning to me: they give *me a discount*.

With a finite verb (like hagan), it usually goes before the verb:

• me hagan = give me

If you used an infinitive, it could attach:

• puede que vayan a hacerme un descuento (less direct, but valid)

La multa suggests a specific fine that is already known in context: the fine (I got).
Una multa would mean a fine in a more general or indefinite sense.

In real life, if you’re talking about the fine you received, la multa is the natural choice.

Yes: puede is 3rd person singular, and the subject is an implicit it (like English it may be that...). Spanish often omits that dummy subject.

So it’s essentially: (Ello) puede que... but you don’t say ello in normal speech.

Yes. You can say:

• Puede que me hagan un descuento si pago la multa hoy.

Meaning stays basically the same. Putting the si clause first often highlights the condition as the setup; putting it last can sound slightly more like an “afterthought” condition, depending on intonation.