Halkan beszélek, hogy a gyerek tovább aludjon.

Because halk is the adjective quiet/soft, while halkan is the adverb quietly/softly.

Hungarian often forms adverbs from adjectives with -an / -en:

• halk → halkan
• gyors → gyorsan
• lassú → lassan

Since the sentence describes how the speaker is speaking, Hungarian uses the adverb: halkan.

Beszélek means I speak / I am speaking.

It is the 1st person singular present tense form of beszélni (to speak). The ending -ek here shows I.

So:

• beszélek = I speak
• beszélsz = you speak
• beszél = he/she speaks

Hungarian usually does not need the pronoun én (I), because the verb ending already shows the subject.

Because Hungarian normally drops subject pronouns when they are clear from the verb.

In beszélek, the ending already tells you the subject is I, so én is unnecessary.

You could say Én halkan beszélek..., but that would usually add emphasis, something like:

• I’m the one speaking softly...
• As for me, I speak softly...

Without emphasis, the pronoun is usually omitted.

Here hogy introduces a purpose clause: it means so that or in order that.

So the structure is:

main clause + hogy + subordinate clause

In this sentence, the second clause gives the purpose of speaking softly.

This is very common in Hungarian:

• Korán lefekszem, hogy kipihenjem magam.
• Eljöttem, hogy segítsek.

In everyday English, we often use to or so that, but Hungarian frequently uses hogy plus a verb form like aludjon.

Because in Hungarian, subordinate clauses are normally separated by a comma, and a clause introduced by hogy is no exception.

So:

• Halkan beszélek, hogy...

This is standard Hungarian punctuation.

English is often less strict about commas in similar structures, so this is something learners notice quickly.

Because a gyerek means the child.

Hungarian uses the definite article a / az much like English uses the:

• a gyerek = the child
• egy gyerek = a child

Here the child is understood as a specific one, so a gyerek is natural.

Also, a is used before consonant sounds, while az is used before vowel sounds:

• a gyerek
• az ember

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

So aludjon here means something like:

• may sleep
• should sleep
• keep sleeping in the sense required by the purpose clause

It is not a command to the child in the usual direct sense. Instead, it expresses the intended result of the main action.

Compare:

• A gyerek alszik. = The child is sleeping.
  (plain statement)
• ...hogy a gyerek aludjon. = ...so that the child may sleep / so that the child sleep.
  (purpose)

And aludna is conditional (would sleep), which is a different meaning.

It comes from the verb aludni (to sleep).

Aludjon is the 3rd person singular imperative/subjunctive form.

A useful way to think about it is:

• infinitive: aludni
• dictionary idea: to sleep
• purpose-clause form: aludjon

This form is also used for direct commands in the 3rd person:

• Aludjon a gyerek! = Let the child sleep!

But in your sentence, it is not a command by itself; it is part of the hogy clause expressing purpose.

Here tovább means longer, further, or more naturally in this context, continue to.

With aludjon, it gives the idea of keep sleeping or sleep on.

So tovább aludjon means the child should continue sleeping, not just fall asleep.

This is a very common use of tovább:

• dolgozik tovább = keeps working / works on
• megy tovább = goes on / continues
• aludjon tovább = keep sleeping / sleep on

Not with exactly the same meaning.

• tovább aludjon focuses on continuation: the child should keep sleeping
• még aludjon would suggest sleep a bit more / still sleep, which is close, but not quite the same

In this sentence, tovább is the most natural choice if the point is that the child is already asleep and the speaker wants that to continue.

This is a very natural neutral word order.

The first clause starts with halkan, which puts attention on how the speaker is speaking. That makes good sense here, because the manner of speaking is important.

The second clause is also neutral:

• a gyerek = subject
• tovább = adverb
• aludjon = verb

Hungarian word order is flexible, but changes emphasis. For example:

• Beszélek halkan, hogy a gyerek tovább aludjon.
  Also possible, a bit less focused on quietly
• Hogy a gyerek tovább aludjon, halkan beszélek.
  This puts the purpose first: So that the child keeps sleeping...

So the original sentence is a very natural, standard way to say it.

Yes, gyerek is the normal everyday word for child/kid.

Gyermek is also correct, but it sounds more formal, careful, or literary.

So:

• a gyerek = the usual everyday choice
• a gyermek = more formal or elevated

In ordinary speech, gyerek is extremely common and completely natural here.

It can mean either, depending on context.

Hungarian present tense often covers both:

• I speak
• I am speaking

In this sentence, the context strongly suggests a current action: the speaker is talking softly right now for a reason.

So even though Hungarian uses simple present, English may choose a progressive translation depending on context.