Önéletrajzzal jelentkezem az állandó állásra.

Because the suffix -val/-vel changes form after many final consonants.

The base word is önéletrajz (CV / résumé).
When you add -val (with), the v usually assimilates to the final consonant of the word:

• kéz + vel → kézzel
• ház + val → házzal
• önéletrajz + val → önéletrajzzal

So önéletrajzzal means with a CV / using a CV.

Here it means with or by means of.

So önéletrajzzal jelentkezem means something like:

• I apply with a CV
• I am applying by submitting a CV

In Hungarian, -val/-vel can show:

• accompaniment: a barátommal = with my friend
• instrument: késsel = with a knife
• means/manner: önéletrajzzal = with a CV

In this sentence, it is about the means of applying.

Önéletrajz means CV or résumé.

It is made up of:

• ön = self
• élet = life
• rajz = drawing / sketch

So literally it is something like self-life-sketch, but in real usage it simply means CV / résumé.

Jelentkezem is the 1st person singular present tense form of jelentkezik.

So it means:

• I apply
• I am applying
• depending on context, sometimes I sign up or I register

The ending -em here means I.

Because jelentkezik is an -ik verb, its I form is jelentkezem, not jelentkezek in standard usage for learners.

No. Jelentkezik has several related meanings, depending on context:

• to apply for something
  
  • állásra jelentkezik = apply for a job
• to sign up / register
  • tanfolyamra jelentkezik = sign up for a course
• to report / make oneself known
  • jelentkezik a tanárnál = report to the teacher

In your sentence, because of állásra (for a job/position), it clearly means to apply.

Because the verb jelentkezik normally goes with -ra/-re when it means apply for or sign up for something.

So:

• állásra jelentkezik = apply for a job
• egyetemre jelentkezik = apply to a university
• versenyre jelentkezik = enter / sign up for a competition

This is one of those verb + case patterns that you simply need to learn.

Even though -ra/-re often literally means onto, with some verbs it has a more abstract meaning.

Hungarian has two forms of the definite article:

• a before a consonant
• az before a vowel

Since állandó begins with a vowel, you use az:

• az állandó állásra
• compare: a jó állásra

This is just like the difference between a and an in English, though the Hungarian rule is based only on sound, not spelling tradition.

They do look similar, but they mean different things:

• állandó = permanent / constant
• állás = job / position

So állandó állás means permanent job / permanent position.

They are historically related to the idea of standing, but as a learner, it is best to treat them as separate vocabulary items.

Yes, it is understandable and natural enough in many contexts, meaning a permanent position.

Depending on context, Hungarians might also say things like:

• állandó munka
• állandó munkahely
• határozatlan idejű állás (more formal, especially in employment language)

But állandó állás is perfectly clear and normal.

Not necessarily. Hungarian often leaves possession unspoken when it is obvious from context.

So önéletrajzzal jelentkezem can naturally mean:

• I’m applying with a CV
• and in practice often implies with my CV

If you want to make my explicit, you can say:

• az önéletrajzommal jelentkezem

That means I am applying with my CV.

Both are possible; the version without the possessive is more general.

Hungarian word order is flexible and often shows topic or emphasis, not just grammar.

Starting with Önéletrajzzal puts some attention on the means of applying:

• Önéletrajzzal jelentkezem az állandó állásra.
  • roughly: I’m applying for the permanent position with a CV.

If you changed the order, the basic meaning would stay similar, but the emphasis could shift.

For example:

• Az állandó állásra jelentkezem önéletrajzzal.
  • stronger focus on the permanent position
• Jelentkezem az állandó állásra önéletrajzzal.
  • more neutral in some contexts, though the exact feel depends on intonation and situation

So the first word is not random; it helps organize what the speaker is highlighting.

Yes. Hungarian present tense often covers both:

• I apply
• I am applying

So jelentkezem can mean either simple present or present progressive, depending on context.

Hungarian does not normally need a separate form like English am applying.

Yes, that is a very good way to understand it.

The sentence suggests that the speaker is applying using a CV / résumé. In natural English, that could be phrased as:

• I am applying for the permanent position with my CV.
• I am applying for the permanent position by submitting a résumé.

So the Hungarian wording is natural even if the most idiomatic English translation may vary a little.