Annak érdekében, hogy az ügyintézési határidő rövid legyen, sürgősségi eljárást kérünk.

Yes. Annak érdekében, hogy is a common formal expression meaning in order that, so that, or for the purpose of.

You should learn it as one chunk:

annak érdekében, hogy + clause

It is especially common in official, legal, and administrative Hungarian. In everyday speech, people often use simpler alternatives such as azért, hogy or just hogy, depending on the sentence.

This is a useful set phrase, but its structure is also understandable:

• annak = of that / for that
• érdekében = roughly in the interest of it

So literally, the phrase is something like in the interest of that. When followed by hogy, it becomes a purpose expression:

annak érdekében, hogy... = in order that...

Even though it has an internal structure, learners usually do best by memorizing the whole phrase as a formal connector.

Because after hogy in a purpose clause, Hungarian very often uses the imperative/subjunctive-type form.

Here:

• legyen = 3rd person singular form of lenni used for wishes, aims, purpose, or desired results

So:

hogy az ügyintézési határidő rövid legyen
means literally something like
that the processing deadline be short

This sounds strange in normal modern English, but it is completely natural in Hungarian.

So the logic is:

• main action: we request an urgent procedure
• intended result: so that the processing deadline be short

It is a compound expression:

• ügyintézés = administration, handling of a case, processing
• ügyintézési = related to administration/case handling
• határidő = deadline, time limit

So ügyintézési határidő means something like:

• administrative processing deadline
• time limit for handling the case
• processing time limit

In this sentence, time limit is often a more accurate feeling than plain deadline.

The suffix -i often turns a noun into an adjective meaning related to or of something.

Here:

• ügyintézés → ügyintézési = related to case handling/administration
• sürgősség → sürgősségi = urgent / emergency / expedited-related

So this -i suffix is very common in formal Hungarian, especially in compound expressions.

Because Hungarian often uses the definite article where English may or may not use the.

Here, az ügyintézési határidő refers to a specific, understood processing time limit connected with the case or procedure. So the definite article is natural.

In official Hungarian, using the article in this kind of phrase is very normal.

sürgősségi eljárás means urgent procedure, expedited procedure, or in some contexts emergency procedure.

In this sentence, expedited procedure is probably the best sense.

The -t on eljárást is the accusative ending, which marks the direct object of the verb.

• base form: eljárás = procedure
• object form: eljárást = procedure, as the thing being requested

Since kérünk means we request, the thing requested must be in the accusative.

This is a very good question, because it involves Hungarian’s indefinite vs. definite conjugation.

• kérünk = we request with an indefinite object
• kérjük = we request it / we request the ... with a definite object

Here the object is:

sürgősségi eljárást

There is no definite article before it, so it is treated as indefinite. That is why the verb is kérünk.

If the sentence had a definite object, for example:

A sürgősségi eljárást kérjük.

then kérjük would be expected.

Yes. Hungarian word order is flexible, and different orders can shift emphasis.

This sentence begins with the purpose clause:

Annak érdekében, hogy az ügyintézési határidő rövid legyen, ...

That puts the goal first: in order to keep the processing time short...

Then comes the main clause:

sürgősségi eljárást kérünk

You could also say:

Sürgősségi eljárást kérünk annak érdekében, hogy az ügyintézési határidő rövid legyen.

That version is also grammatical. It simply places the main action first.

Yes, very formal.

Several things make it sound official or bureaucratic:

• annak érdekében, hogy
• ügyintézési határidő
• sürgősségi eljárás

This is the kind of Hungarian you might see in:

• official letters
• applications
• legal or administrative documents
• formal requests to authorities

In everyday speech, people would usually choose simpler wording.

Probably not in this exact form, unless they were intentionally sounding official.

A more everyday-style sentence might be something like:

Azért kérünk sürgősségi eljárást, hogy rövid legyen az ügyintézés határideje.

or even more naturally in some contexts:

Azért kérünk sürgősségi eljárást, hogy gyorsabb legyen az ügyintézés.

The original sentence is perfectly correct, but it clearly belongs to a formal register.

Kérünk means we request.

So the speaker is either:

• speaking for a group, or
• writing in a formal context on behalf of more than one person

If a single person were saying the same thing, it would normally be:

kérek = I request

So:

• kérek = I request
• kérünk = we request