A diák levest kér, mellé pedig egy pohár vizet.

Because they are direct objects of the verb kér (to ask for).

In Hungarian, a direct object usually takes the accusative ending, most often -t:

• leves → levest
• víz → vizet

So this is the Hungarian way of marking what the student is asking for.

Because Hungarian does not always add -t in a completely mechanical way. Some nouns change shape a little before suffixes.

With víz (water), the stem used before endings is viz-, and a vowel appears before -t, giving:

• víz → vizet

So this is just the correct suffixed form you have to learn. Hungarian has quite a few nouns like this, where the base form and the suffixed form are not perfectly predictable from English.

Hungarian verbs can use two different conjugations with transitive verbs:

• indefinite conjugation
• definite conjugation

Here the things being asked for are indefinite/non-specific:

• levest = soup
• egy pohár vizet = a glass of water

So the verb is kér.

If the object were definite, you would typically get kéri instead:

• A diák a levest kéri. = The student is asking for the soup.

This is a very important Hungarian pattern.

Because with food and drink, Hungarian often leaves out the indefinite article when talking about what someone orders, eats, drinks, or asks for.

So:

• levest kér = asks for soup

This sounds natural, just like English often uses mass nouns without a/an.

You might see egy levest in some contexts, but that would usually sound more like one soup / a soup portion / one kind of soup, depending on the situation. In a simple ordering sentence, levest kér is very natural.

Here mellé means something like:

• with it
• alongside it
• on the side

Literally, it has the sense of to beside it / next to it.

In this sentence, the thing it refers to is understood from context: the soup.

So:

• mellé egy pohár vizet
  roughly = a glass of water with it

This is a very natural Hungarian way to add an accompanying item.

Pedig is a connective word with several uses, depending on context. Here it gives the feeling of:

• and
• and additionally
• while/as for
• and with that

In this sentence, it helps introduce the extra item in a slightly more elegant or contrastive way than plain és.

So the second part feels like:

• and with that, a glass of water
• and alongside it, a glass of water

It is not easy to translate with one fixed English word every time; its job is often more about tone and connection than literal meaning.

Because in Hungarian measure/container expressions often work differently from English.

Here the whole phrase means a glass of water, but since the phrase is functioning as the object of kér, Hungarian commonly marks the substance noun with the accusative:

• egy pohár vizet

This pattern is very common:

• két üveg sört = two bottles of beer
• három kiló almát = three kilos of apples
• egy csésze teát = a cup of tea

You may also meet egy pohár víz in other grammatical contexts, for example as a naming phrase:

• Ez egy pohár víz. = This is a glass of water.

But after kér, egy pohár vizet is the natural form.

Because Hungarian word order is much more flexible than English word order.

Placing the indefinite object right before the verb is very common and natural:

• A diák levest kér.

This does not mean Hungarian is simply “object-verb” all the time. Rather, word order often reflects information structure and emphasis.

In this sentence, levest kér forms a natural unit: asks for soup.

So for an English speaker, the main thing to remember is:

• Hungarian word order is not as rigid as English.
• The position before the verb is often important.

Yes, in a way. The verb kér is understood again, but it is not repeated because that would be unnecessary.

So the full expanded version would be something like:

• A diák levest kér, mellé pedig egy pohár vizet kér.

But Hungarian often leaves out repeated material when the meaning is clear. This is very natural in short everyday sentences.

So the actual sentence is efficient and idiomatic, not incomplete.

A is the Hungarian definite article (the) before a consonant.

• a diák = the student

The form az is used before vowel sounds:

• az alma = the apple

So here a is simply the correct form of the definite article.

Depending on context, a diák may refer to:

• a specific student already known in the conversation, or
• a typical/student character in an example sentence

But grammatically, it is the definite article.

Yes, és would be possible, but it would not feel exactly the same.

For example:

• A diák levest kér, és egy pohár vizet is.

That is possible, but mellé pedig gives a more natural with it / on the side feeling. It makes the water sound like an accompaniment to the soup, not just another random thing on the list.

So pedig is helping shape the relationship between the two requested items.