Min mor vil helst undgå motorvejen, hvis der er meget vejarbejde og lange køer.

Motorvejen is the definite form: the motorway.

In Danish, the definite article is often added to the end of the noun:

• en motorvej = a motorway
• motorvejen = the motorway

So Danish usually does not need a separate word for the in cases like this.

Because it comes after vil.

After modal verbs such as:

• vil = want to / will
• kan = can
• skal = must / shall
• må = may / must

the next verb is normally in the infinitive without at.

So:

• vil undgå = wants to avoid
• not vil at undgå

This is similar to English wants to avoid, except Danish leaves out the equivalent of to after modal verbs.

Helst means something like preferably, rather, or would rather.

In Danish, adverbs like this often come after the finite verb, so:

• vil helst undgå = would preferably avoid / would rather avoid

This is a very natural placement.
Compare:

• Jeg vil helst blive hjemme. = I would rather stay home.

So helst modifies the whole idea of what she wants to do.

Because vil is a modal verb, and modal verbs are followed directly by the infinitive.

So:

• hun vil undgå = she wants to avoid
• hun kan køre = she can drive
• hun skal arbejde = she must work

But after many other verbs, Danish does use at:

• Hun prøver at undgå motorvejen. = She tries to avoid the motorway.

So the lack of at is specifically because of vil.

Because in a subordinate clause introduced by hvis, Danish keeps normal clause order.

A basic Danish clause often has:

• subject + verb

Here:

• der = subject-like element
• er = verb

So:

• hvis der er meget vejarbejde = if there is a lot of roadwork

You do not invert the verb and subject after hvis.
Inversion is common in main clauses after certain elements, but not here.

In Danish, der er is the normal way to say there is / there are.

So:

• der er meget vejarbejde = there is a lot of roadwork
• der er lange køer = there are long queues

This is different from det er, which usually means it is.

Compare:

• Der er en bil udenfor. = There is a car outside.
• Det er en bil. = It is a car.

So here der er is exactly what Danish normally uses for existence.

Because the two nouns behave differently.

1. vejarbejde

This is treated as an uncountable noun here, like roadwork in English.

So Danish uses:

• meget = much / a lot of

Example:

• meget vejarbejde = a lot of roadwork

2. køer

This is a plural countable noun, so Danish uses a plural adjective:

• lange køer = long queues

So the difference comes from:

• vejarbejde = uncountable
• køer = plural countable

Because køer is plural, and adjectives usually take -e in the plural.

So:

• en lang kø = a long queue
• lange køer = long queues

This -e ending is very common with plural nouns.

You can think of it as adjective agreement:

• singular common gender: often basic form
• plural: often -e

Yes, køer can mean either queues or cows.

That is because it can be the plural of two different words:

• en ko → køer = cow → cows
• en kø → køer = queue → queues

The meaning comes from context.
In a traffic sentence with motorvejen and vejarbejde, it clearly means queues.

Hvis means if and is used for a condition.

Here the idea is conditional:

• she would rather avoid the motorway if there is a lot of roadwork and long queues

Når usually means when, especially when something is expected or habitual.

So:

• hvis = if
• når = when

If you changed it to når, the meaning would shift from a condition to something more like an expected situation.

Helst often expresses preference.

In this sentence, vil helst undgå means something like:

• would rather avoid
• prefers to avoid

It is stronger than just vil undgå, because it adds the idea of preference.

Compare:

• Min mor vil undgå motorvejen. = My mother wants to avoid the motorway.
• Min mor vil helst undgå motorvejen. = My mother would rather avoid the motorway.

Because hvis der er meget vejarbejde og lange køer is a subordinate clause.

In standard Danish punctuation, a subordinate clause is usually separated from the main clause with a comma.

So the structure is:

• main clause: Min mor vil helst undgå motorvejen
• subordinate clause: hvis der er meget vejarbejde og lange køer

That is why the comma appears before hvis.

It can often do either, depending on context.

The definite form motorvejen literally means the motorway, but in everyday Danish it may refer to:

• a specific motorway already understood from context, or
• the motorway route that would normally be taken

So even though it is definite, it does not always have to mean one uniquely identified motorway in a very strict sense. It can simply mean the motorway option that both speaker and listener understand.