Espero que tu garganta mejore pronto.

Because mejore is the present subjunctive form of mejorar.

After esperar que, Spanish normally uses the subjunctive when you are expressing a hope, wish, or expectation about something that is not presented as a fact.

• Espero que tu garganta mejore pronto = I hope your throat gets better soon.
• Tu garganta mejora pronto would sound wrong here, because mejora is not the form you need after espero que.

So the pattern is:

• Espero que + subjunctive

Examples:

• Espero que estés bien.
• Espero que descanses.
• Espero que te mejores pronto.

Because que introduces the second clause: the thing you are hoping for.

In this sentence, there are really two parts:

• Espero = I hope
• que tu garganta mejore pronto = that your throat gets better soon

So que works like that in English, even though English often leaves that out:

• I hope (that) your throat gets better soon.

In Spanish, you normally must include que here.

Because here tu is the possessive adjective meaning your.

Spanish distinguishes between:

• tu = your
• tú = you

So in tu garganta, it means your throat, not you throat.

Compare:

• Tu garganta = your throat
• Tú estás cansado = you are tired

Spanish often uses the definite article with body parts, especially when the owner is already clear from a reflexive pronoun:

• Me duele la garganta. = My throat hurts.
• Te duele la cabeza. = Your head hurts.

But in this sentence, the speaker is directly referring to your throat as the subject of the clause, so tu garganta is completely natural.

Both body-part patterns exist in Spanish, but they are used in different structures. Here, tu garganta is straightforward and clear.

No. Garganta means throat, while cuello means neck.

So:

• garganta = throat, especially the inside area that can hurt when you are ill
• cuello = neck, the outside part connecting the head and body

If someone has a sore throat, garganta is the right word.

Examples:

• Me duele la garganta. = My throat hurts.
• Me duele el cuello. = My neck hurts.

Because Spanish often omits subject pronouns when the verb form already makes the meaning clear enough.

Here, the subject is tu garganta, so there is no need to add a pronoun.

Spanish prefers:

• Espero que tu garganta mejore pronto.

Not:

• Espero que ella mejore pronto, unless ella refers to some previously mentioned feminine noun and you want to avoid repeating it.

Since tu garganta is already stated, adding a pronoun would be unnecessary.

Not exactly.

In this sentence, pronto means soon, not quickly.

• pronto = soon
• rápidamente = quickly

So:

• Espero que tu garganta mejore pronto = I hope your throat gets better soon.
• Espero que tu garganta mejore rápidamente = I hope your throat gets better quickly.

Sometimes soon and quickly are similar in idea, but they are not the same meaning.

Yes. Spanish adverbs often have some flexibility.

You can say:

• Espero que tu garganta mejore pronto.
• Espero que pronto mejore tu garganta.
• Espero que tu garganta pronto mejore.

But the most natural and common version here is:

• Espero que tu garganta mejore pronto.

That word order sounds the most neutral and idiomatic.

Yes, and it is very common.

There is a small difference in focus:

• Espero que tu garganta mejore pronto focuses specifically on your throat
• Espero que te mejores pronto focuses on you getting better

So if the person has a sore throat but is otherwise okay, tu garganta mejore makes sense because it refers to that specific problem. If the person is generally unwell, te mejores pronto is broader.

In everyday conversation, many people would probably say:

• Espero que te mejores pronto.

Because the speaker is expressing a hope right now.

• Espero = I hope

It is the present indicative of esperar.

So the structure is:

• Espero = main verb in the present
• que tu garganta mejore = subordinate clause in the present subjunctive

This combination is very common in Spanish:

• Espero que venga.
• Espero que salga bien.
• Espero que descanses.

Yes, it is natural and correct in Spain.

That said, in everyday Spanish from Spain, people might also commonly say:

• Espero que te mejores pronto.
• Que te mejores pronto.
• Espero que se te pase pronto. if referring to the discomfort or illness passing

Your original sentence is perfectly understandable and natural, especially if you want to mention the throat specifically. It sounds caring and polite.