Min søster siger, at hendes mand nok skal hente børnene i dag.

At introduces a subordinate clause, like English that.

So Min søster siger, at ... is literally My sister says that ...

In English, that is often optional, but in Danish at is very common and usually natural to include. In informal speech it can sometimes be omitted, but learners should normally keep it.

Because after at, Danish uses subordinate-clause word order.

A useful pattern is:

subject + sentence adverb + finite verb + infinitive/object

Here:

• hendes mand = subject
• nok = sentence adverb
• skal = finite verb
• hente = infinitive

So you get:

at hendes mand nok skal hente børnene i dag

If this were a main clause, the order would be:

Hendes mand skal nok hente børnene i dag.

That contrast is very important in Danish grammar.

Here nok does not mean enough.

In this sentence, nok is an adverb meaning something like:

• probably
• likely
• sometimes surely or don’t worry, it will happen

With skal, the combination skal nok often has a reassuring tone. It can mean not just future, but also confidence that the action will happen.

So the feeling is often closer to:

• will probably
• will surely
• will take care of it

The exact nuance depends on context and tone.

Not exactly.

Skal can mean several things in Danish:

• obligation: must / has to
• plan or arrangement: is going to
• future meaning: will
• promise or assurance, especially with nok

In this sentence, skal does not sound like a strong must. With nok, it sounds more like a future plan or reassurance:

her husband will / is going to pick up the children today

So skal here is closer to will than to must.

Because sin/sit/sine can only refer back to the subject of the same clause.

In this sentence, min søster is the subject of the main clause:

Min søster siger ...

But after at, you start a new clause:

at hendes mand nok skal hente børnene i dag

Inside that clause, sin cannot reach back to min søster, so Danish uses hendes.

A useful comparison:

• Min søster elsker sin mand = My sister loves her own husband
  Here sin works because it refers to the subject of the same clause: min søster.

• Min søster siger, at hendes mand kommer
  Here you need hendes, because mand is inside a new clause after at.

This is one of the biggest differences between Danish and English possessives.

It can mean either, depending on context.

• en mand usually means a man
• hendes mand usually means her husband

When mand follows a possessive like min, hans, or hendes, it very often means husband rather than just man.

Because Danish usually marks definiteness with an ending instead of a separate word.

Here is the pattern:

• et barn = a child
• børn = children
• børnene = the children

So -ene is the definite plural ending.

This word is a little irregular because the singular barn changes to børn in the plural.

Not necessarily from grammar alone.

Børnene literally means the children. Danish often leaves possession unstated if it is obvious from context.

So in real life, listeners may understand that these are the sister and her husband’s children, but the sentence itself only says the children, not explicitly their children.

Yes, but the position at the end is the most neutral and natural here.

... hente børnene i dag is a very standard way to say it.

You can move time expressions for emphasis, especially to the front of the whole sentence:

I dag siger min søster, at hendes mand nok skal hente børnene.

That gives extra emphasis to today. But for learners, leaving i dag at the end is a good default.

Because søster is a common-gender noun.

Danish singular possessives change like this:

• min for common gender
• mit for neuter
• mine for plural

So:

• min søster = my sister
• mit barn = my child
• mine børn = my children

The same idea applies throughout Danish noun phrases.

Sometimes in informal speech, yes, but not always, and not as a beginner habit.

Some speakers may say something like:

Min søster siger hendes mand nok skal hente børnene i dag

But the version with at is clearer and more standard:

Min søster siger, at hendes mand nok skal hente børnene i dag.

So for learners, it is best to keep at.