A fogszuvasodás miatt új tömés kell.

Miatt means because of / due to. It is a postposition, so in Hungarian it comes after the noun phrase, not before it as in English.

So:

• a fogszuvasodás miatt = because of the tooth decay

The noun before miatt stays in its basic form; you do not add a special case ending to it here.

A is the definite article, meaning the.

Hungarian often uses the definite article in places where English may or may not use one, especially when talking about a specific, known condition or problem. Here it suggests the tooth decay in question, not tooth decay as an abstract idea in general.

So a fogszuvasodás miatt sounds like because of the tooth decay / because of that tooth decay problem.

In very short or note-like style, you may also see the article omitted, but the version with a is very natural.

It is a compound word:

• fog = tooth
• szuvas = decayed / carious
• szuvasodás = decay / the process of becoming decayed

So fogszuvasodás literally means something like tooth-decay, i.e. tooth decay or dental caries.

Usually fogszuvasodás refers to tooth decay as a condition or problem, rather than one single hole.

In everyday English, people often say cavity when they mean one bad spot in a tooth. Hungarian can sometimes talk more broadly with fogszuvasodás. In casual speech, a single cavity may also be described as a lyuk a fogban (a hole in the tooth).

So in this sentence, fogszuvasodás is best understood as tooth decay as the reason a filling is needed.

Here tömés means a dental filling.

The word comes from the verb töm, which means to fill / stuff / pack. So tömés can mean filling in different contexts, but in a sentence about teeth, it clearly means a tooth filling.

Because kell does not require a direct object here.

In this kind of sentence, the thing that is needed appears in the basic form (nominative), not in the accusative.

So:

• új tömés kell = a new filling is needed

not

• új tömést kell in this structure

English speakers often expect an object because they think of need as taking one, but Hungarian is structuring this more like a new filling is necessary / is needed.

Because Hungarian often leaves out the indefinite article in this kind of need/requirement statement.

So:

• új tömés kell = a new filling is needed

This is very natural.

If you say:

• egy új tömés kell

that is also possible, but it sounds a bit more like you are emphasizing one new filling.

So the version without egy is more neutral here.

Here kell means is needed / is necessary.

It is a very common Hungarian verb that can express ideas like:

• must
• need
• have to
• be necessary

In this sentence, the most natural understanding is:

• új tömés kell = a new filling is needed

So it is not really about obligation in the sense of must; it is more about necessity.

It is not there explicitly.

Hungarian often says what is needed without naming the person, especially in medical or practical contexts. From the situation, it is obvious that it means something like:

• You need a new filling
• A new filling is needed

So English often adds you, but Hungarian does not have to.

Hungarian word order is flexible and often shows topic and focus, not just subject-verb-object order.

Here:

• A fogszuvasodás miatt sets the scene: because of the tooth decay
• új tömés is the important new information
• kell comes after that

So the sentence presents the cause first, then says what is needed.

This is a very natural order for giving information in context.

Yes. For example:

• Új tömés kell a fogszuvasodás miatt.
  Same basic meaning, just a different information flow.

• A fogszuvasodás miatt egy új tömés kell.
  Also possible; egy makes one new filling a bit more explicit.

• A fogszuvasodás miatt tömésre van szükség.
  A more formal way to say A filling is necessary because of the tooth decay.

So the original sentence is natural, but it is not the only possible wording.