Jag hoppas att banken godkänner ansökan och inte avvisar den igen.

Because att introduces the clause that tells us what the speaker hopes.

• Jag hoppas = I hope
• att banken godkänner ansökan ... = that the bank approves the application ...

In English, that is often omitted: I hope the bank approves...
In Swedish, att is much more commonly kept here, so Jag hoppas att ... is the normal pattern.

Banken is the definite form, meaning the bank.

• en bank = a bank
• banken = the bank

The sentence is talking about a specific bank, not just any bank.

Swedish often uses the present tense where English might use either present or future.

So:

• banken godkänner literally looks like the bank approves
• but in context it can naturally mean the bank approves / will approve

After verbs like hoppas (hope), this is very normal in Swedish. You do not need ska here.

They are present-tense forms of the infinitives:

• godkänna = to approve
• avvisa = to reject / turn down
• godkänner = approves
• avvisar = rejects

So the structure is:

• banken godkänner = the bank approves
• banken avvisar = the bank rejects

Because Swedish, like English, often leaves out the repeated subject when two verbs share the same subject.

So this:

• att banken godkänner ansökan och inte avvisar den igen

means:

• that the bank approves the application and does not reject it again

The subject banken belongs to both verbs:

• banken godkänner
• (banken) inte avvisar

Here, inte is negating only the second action:

• godkänner ansökan och inte avvisar den igen
• approves the application and does not reject it again

So inte is placed right before the second verb phrase to show that only avvisar den igen is negative.

Compare:

• Jag hoppas att banken inte avvisar den igen.
  = I hope that the bank doesn’t reject it again.

In your sentence, the idea is more specifically:

• I hope it approves it, and not rejects it again.

Because ansökan is an en-word.

In Swedish:

• en-word singular nouns are referred to with den
• ett-word singular nouns are referred to with det

So:

• en ansökan = an application
• den = it referring back to ansökan

That is why den is correct here.

This is a very common question. Ansökan is one of those nouns whose indefinite singular and definite singular are spelled the same.

So depending on context:

• en ansökan = an application
• ansökan = the application

In this sentence, it usually means the application, because the speaker is talking about a specific application that has already been mentioned or is understood from context.

Igen means again.

So:

• avvisar den igen = rejects it again

Putting igen at the end is very natural in Swedish. It often comes late in the clause, especially after the object pronoun.

This word order sounds normal and idiomatic.

They are opposites in meaning here:

• godkänna = approve
• avvisa = reject / turn down

So the sentence contrasts two possible outcomes:

• the bank approves the application
• the bank does not reject it again

Yes, sometimes. This is a useful vocabulary question.

• avslå en ansökan often means to formally reject/deny an application
• avvisa can also mean reject, but it can sometimes sound a bit broader, like dismiss, turn away, or refuse

For official applications, avslå is often very common. But avvisa is still understandable and can work depending on context.

In careful speech, jag is pronounced roughly like yahg.
In everyday spoken Swedish, it is very often reduced to something like ya.

So:

• careful: jag hoppas
• common spoken: ja hoppas

Both are useful to recognize when listening.