Ef hún hafnar þessu starfi, sækir hún um annað starf strax.

Hafna is the infinitive, meaning to reject or to decline.

In the sentence, the verb has to match hún (she), so it appears in the 3rd person singular present tense:

• ég hafna = I reject
• þú hafnar = you reject
• hún hafnar = she rejects

So Ef hún hafnar þessu starfi means If she rejects this job.

Because the verb hafna governs the dative case in Icelandic.

So the thing being rejected is not in the nominative or accusative here, but in the dative:

• þetta starf = this job (nominative/accusative)
• þessu starfi = this job (dative)

Both words change because they must agree in case, gender, and number:

• þessu = dative singular neuter of þessi
• starfi = dative singular of starf

So after hafna, you say:

• hafna þessu starfi = reject this job

Because sækja um takes the accusative case.

So while hafna uses the dative, sækja um uses the accusative:

• hafna þessu starfi → dative
• sækja um annað starf → accusative

Here:

• annað = accusative singular neuter of annar
• starf = accusative singular neuter

So annað starf means another job in the form required by sækja um.

For learners, it is best to treat sækja um as a fixed expression meaning to apply for.

Literally, sækja can have other meanings in other contexts, but in this construction:

• sækja um starf = apply for a job
• sækja um skóla = apply for school
• sækja um leyfi = apply for permission

So yes, um is a preposition, but sækja um functions together as the normal Icelandic way to say apply for.

This is because Icelandic is a verb-second language.

When the sentence begins with the subordinate clause Ef hún hafnar þessu starfi, that whole clause takes the first position. In the following main clause, the finite verb then comes next:

• Ef hún hafnar þessu starfi, sækir hún um annað starf strax.

So the structure is roughly:

1. first element: Ef hún hafnar þessu starfi
2. second position: sækir
3. then the subject: hún

If the main clause stood alone, you would normally say:

• Hún sækir um annað starf strax.

But after the if-clause comes first, Icelandic switches to:

• Ef ..., sækir hún ...

Because Icelandic often uses the present tense for future meaning when the context makes the time clear.

That is very common in conditional sentences with ef (if):

• Ef hún hafnar þessu starfi, sækir hún um annað starf strax.

Even though both verbs are present tense grammatically, the meaning is naturally future:

• If she rejects this job, she’ll apply for another one right away.

English also sometimes does something similar in the if-clause:

• If she rejects this job, ...

So this is normal Icelandic usage, not a mistake or a special tense you are missing.

Not here. Ef means if, and in this sentence the verbs are in the indicative, which is normal for a real or likely condition.

So:

• Ef hún hafnar þessu starfi ... = If she rejects this job ...

This is a straightforward condition, not something especially hypothetical or contrary to fact.

In other contexts, Icelandic can use different moods, but in a sentence like this, the present indicative is exactly what you would expect.

Strax means immediately, right away, or at once.

In this sentence:

• sækir hún um annað starf strax
• she applies for another job right away

Putting strax at the end is very natural. Adverbs like this are somewhat flexible in Icelandic, but sentence-final position is common and clear.

The difference is caused by case.

It is the same noun, starf (job), but different verbs require different cases:

• hafna þessu starfi → dative
• sækja um annað starf → accusative

So:

• starfi = dative singular
• starf = accusative singular

This is a very common feature of Icelandic: the form of the noun changes depending on what verb or preposition it goes with.

Normally, it should be repeated.

Icelandic usually states the subject in each clause unless there is a special reason not to. So:

• Ef hún hafnar þessu starfi, sækir hún um annað starf strax.

This is the natural full sentence. Omitting the second hún would not sound like standard neutral Icelandic here.

So even though English also repeats she, it is worth noticing that Icelandic does the same in this structure.