Når børnene sover, kan jeg læse videre i romanen.

Når introduces a time clause. Here it means when.

So Når børnene sover means when the children are sleeping / when the children sleep.

A useful detail:

• når can mean when in a general or repeated sense, like whenever
• in many contexts, it is also used for future time: when the children fall asleep / when the children are asleep

So it is a very common word for time clauses in Danish.

Because barn is irregular in the plural.

The forms are:

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

So børnene is the definite plural form: the children.

This is something you simply have to learn as an irregular noun.

The ending -ene marks the definite plural for many Danish nouns.

So:

• børn = children
• børnene = the children

In Danish, the definite article is often attached to the end of the noun instead of being a separate word.

Compare:

• en roman = a novel
• romanen = the novel

Because sover is the present tense form of the verb sove.

• sove = to sleep
• sover = sleep / are sleeping / is sleeping

In Danish, the present tense is often made by adding -r:

• læse → læser
• sove → sover

So børnene sover means the children are sleeping or the children sleep, depending on context.

This is one of the most important word-order rules in Danish.

The sentence begins with a subordinate clause:

• Når børnene sover

After that comes the main clause:

• kan jeg læse videre i romanen

In a Danish main clause, the finite verb usually comes in second position. This is often called the V2 rule.

So:

• first element: Når børnene sover
• second element: kan
• then the subject: jeg

That is why you get:

• Når børnene sover, kan jeg ...

If the sentence started directly with the subject, it would be:

• Jeg kan læse videre i romanen, når børnene sover.

Both are correct, but the word order changes because of what comes first.

Yes, this comma is normal and expected here.

It separates the opening subordinate clause:

• Når børnene sover

from the main clause:

• kan jeg læse videre i romanen

So the comma helps show the structure of the sentence clearly.

Læse videre means continue reading or read on.

The word videre often gives the idea of:

• continuing
• going further
• carrying on

So:

• læse = read
• læse videre = keep reading / continue reading

It does not mean simply read. It suggests that the speaker has already started the novel and wants to continue.

I romanen means in the novel.

Here it works naturally with læse videre:

• læse videre i romanen = continue reading in the novel

This focuses on continuing at some point inside that book.

Also notice:

• roman = novel
• romanen = the novel

So -en is the definite ending for a common-gender singular noun.

Yes, absolutely.

That version is also correct:

• Jeg kan læse videre i romanen, når børnene sover.

The difference is mostly about emphasis and information flow.

• Når børnene sover, kan jeg læse videre i romanen.
  This starts with the time condition: when the children are asleep...

• Jeg kan læse videre i romanen, når børnene sover.
  This starts with I can continue reading...

Both are natural.

Sometimes, but the meaning changes slightly.

• når = when
• mens = while

So:

• Når børnene sover, kan jeg læse videre i romanen.
  = When the children are asleep, I can continue reading the novel.

• Mens børnene sover, kan jeg læse videre i romanen.
  = While the children are sleeping, I can continue reading the novel.

Mens emphasizes that the two actions happen at the same time.
Når is the more general when.

Kan is the present tense of kunne, meaning can / be able to.

So:

• kan jeg læse videre = I can continue reading

It expresses possibility or ability. In this sentence, it suggests that once the children are asleep, the speaker has the chance or opportunity to read.

It can be either, depending on context.

Danish present tense often covers both:

• simple present: sleep
• present progressive: are sleeping

So børnene sover can mean:

• the children sleep
• the children are sleeping

In this sentence, English would usually prefer are sleeping or are asleep, but Danish simply uses the present tense form sover.