A gyerek egyszerre nevet és köhög, ezért anya vizet hoz neki.

Hungarian has two forms of the definite article:

• a before consonant sounds
• az before vowel sounds

Since gyerek begins with a consonant sound, you say a gyerek = the child.

Examples:

• a ház = the house
• az alma = the apple

Gyerek means child/kid and is very common in everyday speech.

Gyermek also means child, but it sounds more formal, careful, or official.

So:

• gyerek = everyday, natural
• gyermek = more formal

In this sentence, gyerek is the normal conversational choice.

Here egyszerre means at the same time or simultaneously.

So:

• A gyerek egyszerre nevet és köhög
  = The child is laughing and coughing at the same time

Be careful: in other contexts, egyszerre can also suggest all at once or suddenly together, depending on the sentence.

Both are 3rd person singular present tense forms:

• nevet = he/she laughs / is laughing
• köhög = he/she coughs / is coughing

The subject is a gyerek = the child, which is singular, so the verbs are singular too.

Hungarian present tense often covers both simple present and present continuous English meanings, depending on context.

So:

• nevet can mean laughs or is laughing
• köhög can mean coughs or is coughing

És simply means and.

It connects the two actions:

• nevet = laughs
• köhög = coughs

So:

• nevet és köhög = laughs and coughs

Ezért means therefore, so, or because of this.

It connects the first clause to the result in the second clause:

• A gyerek egyszerre nevet és köhög, ezért anya vizet hoz neki.
• The child is laughing and coughing at the same time, so mother brings water to him/her.

It is a very common connector for showing consequence.

In Hungarian, family-role words like anya, apa, nagyi can appear without an article when they are used almost like names inside a family context.

So here:

• anya = mom / mother

This sounds natural, as if from the family’s point of view.

Compare:

• Anya vizet hoz. = Mom brings water.
• Az anya vizet hoz. = The mother brings water.
  This sounds more detached, like an outside description.

And az anyja would mean his/her mother, which is more specific and different in tone.

Because vizet is the accusative form, used for the direct object of the verb.

• víz = water
• vizet = water (as the thing being brought)

In this sentence, water is what mom brings, so it must be in the accusative:

• anya vizet hoz = mom brings water

Notice that the vowel changes:

• víz → vizet

This kind of stem change is normal in Hungarian.

Hungarian verbs often have two conjugations:

• indefinite conjugation
• definite conjugation

Here the object is vizet = water, which is not a specific, clearly definite item like the water. It is more like some water.

So Hungarian uses the indefinite verb form:

• hoz = brings

If the object were definite, you would use the definite conjugation:

• A vizet hozza. = He/she brings the water.

So in your sentence:

• anya vizet hoz neki = mom brings him/her water

Neki means to him, to her, or sometimes for him/her, depending on context.

It is the dative form of the pronoun:

• ő = he/she
• neki = to him/to her

In this sentence:

• anya vizet hoz neki
• mom brings water to him/her

Since a gyerek was already mentioned, neki refers back to the child.

Hungarian does not normally distinguish he and she in the third person singular pronoun.

So:

• ő = he or she
• neki = to him or to her

You understand the gender from context if needed. This is very different from English, where you usually must choose he or she.

Yes, Hungarian word order is flexible, but different orders give different emphasis.

The version here is natural and neutral:

• anya vizet hoz neki

This presents the information straightforwardly: mom brings water to the child.

Other orders are possible, for example:

• Anya neki hoz vizet.
  This can emphasize to him/her.
• Vizet hoz neki anya.
  This may sound more marked or stylistic.

Hungarian word order is strongly connected to focus and emphasis, not just grammar.

Usually hoz means bring.

But depending on context, English may translate it as fetch if the person goes to get something and then brings it back.

So here:

• anya vizet hoz neki

could be understood as:

• mom brings him/her water or naturally in some situations:
• mom fetches some water for him/her

The basic Hungarian verb is still hoz.

Yes, it is natural and understandable.

It gives a simple sequence:

1. The child is laughing and coughing at the same time.
2. Because of that, mom brings water.

A very close English-style natural translation would be:

• The child is laughing and coughing at the same time, so Mom brings him/her some water.

Yes, and in this sentence Mom is often the more natural translation.

• anya literally corresponds to mother/mom
• in family context, it often feels closer to Mom

So depending on tone, you could translate it as:

• therefore mother brings water to him/her
• so Mom brings him/her some water

The second usually sounds more natural in everyday English.