Azért öblítek, hogy tiszta legyen a fogköz.

Here azért ... hogy is a very common Hungarian pattern meaning so that, in order that, or for the purpose of.

So:

• Azért öblítek, hogy tiszta legyen a fogköz. = I rinse so that the gap between the teeth will be clean.

A useful thing to remember is that azért often pairs with hogy to introduce a purpose:

• Azért tanulok, hogy levizsgázzak. = I study so that I pass the exam.

Be careful: azért can also mean for that reason in other contexts, so the full pattern matters.

Öblítek comes from the verb öblít, which means to rinse.

The form öblítek means:

• I rinse
• I am rinsing

Hungarian present tense often covers both simple present and present progressive, depending on context.

So öblítek is:

• verb stem: öblít-
• 1st person singular ending: -ek

Because of vowel harmony and verb type, the ending here is -ek.

This is about the Hungarian indefinite vs. definite conjugation.

• öblítek = indefinite conjugation
• öblítem = definite conjugation

Hungarian uses the definite conjugation when the verb has a definite direct object.

In this sentence, there is no explicit definite direct object after öblítek, so Hungarian uses the indefinite form:

• Azért öblítek... = I rinse...

If you said something like:

• A számat öblítem. = I rinse my mouth.

then öblítem would be natural, because a számat is a definite object.

So the sentence is not saying I rinse it, but more generally I rinse / I do rinsing.

Because after a purpose clause with hogy, Hungarian often uses a form that English learners often think of as subjunctive-like.

legyen is the 3rd person singular imperative/subjunctive form of lenni (to be).

So:

• hogy tiszta legyen literally: so that it be clean naturally: so that it is / will be clean

Hungarian does not use van here:

• not hogy tiszta van
• not normally hogy tiszta lesz

In a purpose clause, legyen is the standard choice.

You will see the same thing in many similar sentences:

• Elmegyek, hogy ott legyek időben. = I’m leaving so that I’ll be there on time.

• Leírom, hogy világos legyen. = I’ll write it down so that it’s clear.

tiszta means clean.

legyen means something like be in a purpose/wish/result sense.

Together:

• tiszta legyen = be clean / will be clean / should be clean

This is a very common Hungarian structure:

• adjective + legyen

Examples:

• fontos legyen = be important
• világos legyen = be clear
• készen legyen = be ready

So in your sentence, the speaker is expressing the goal of rinsing: the interdental space should end up clean.

Fogköz is a compound noun:

• fog = tooth
• köz = space between, gap, middle, interior space

So fogköz literally means tooth-gap or space between teeth.

In natural English, you might translate it as:

• the space between the teeth
• the gap between the teeth
• interdental space

This is the kind of compact compound Hungarian likes to form.

Hungarian often uses the definite article where English might or might not use the, depending on style.

Here a fogköz means:

• the interdental space
• the space between the teeth

The article makes it sound like a specific, understood thing in context.

Also, Hungarian commonly uses the article with body-related or contextually obvious nouns. So this feels natural.

Yes, English often prefers a plural idea here, but Hungarian can use a singular compound noun to refer to the relevant type of space.

So a fogköz does not necessarily mean only one literal gap in a strict mathematical sense. It can refer more generally to the tooth gap area / the space between teeth.

Hungarian often packages ideas differently from English:

• English: between the teeth
• Hungarian: the tooth-gap / interdental space

So the singular is normal and idiomatic.

Both are possible, but they put the focus in slightly different places.

In:

• hogy tiszta legyen a fogköz

the adjective tiszta comes before legyen, which gives prominence to the desired state: clean.

This is very natural in Hungarian when the important point is the result or quality.

If you say:

• hogy a fogköz tiszta legyen

that is also grammatical, and it puts a bit more focus on the interdental space as the thing that should be clean.

So the original sentence is natural and emphasizes the goal-state: clean.

It can mean either, depending on context.

Hungarian present tense does not always distinguish between:

• I rinse
• I am rinsing

So:

• öblítek

could be understood as:

• I rinse
• I’m rinsing
• I do rinsing

In this sentence, because of the purpose clause, English will usually translate it as:

• I rinse so that... or
• I’m rinsing so that...

Both are possible depending on the wider situation.

Yes, you could say:

• Öblítek, hogy tiszta legyen a fogköz.

This still means I rinse so that the interdental space is clean.

Adding azért makes the purpose relationship more explicit and often sounds a bit more deliberate:

• Azért öblítek, hogy... = I’m rinsing for the purpose that...

So azért is not always absolutely required, but it is very common and natural in this pattern.

Fogköz is a real and useful word, but it sounds a bit more specific than very basic everyday vocabulary. You are especially likely to hear it in:

• dental care
• hygiene instructions
• product descriptions
• advice about flossing or rinsing

A learner should recognize it, but in casual everyday conversation people might also express the idea in other ways depending on context.

So the sentence is grammatical and natural, but it has a slightly dental-care / instructional feel.