Hun kunne næsten ikke bevæge håndleddet i går, men i dag kan hun skrive en kort besked.

Because the sentence contrasts yesterday with today.

• kunne = past tense of kan
• kan = present tense

So:

• Hun kunne næsten ikke bevæge håndleddet i går = yesterday, she could hardly move her wrist
• I dag kan hun skrive en kort besked = today, she can write a short message

In other words, the grammar shows a change in her condition from yesterday to today.

This is because Danish normally follows the V2 rule in main clauses: the finite verb must come in the second position.

If the sentence starts with a time expression like i dag, then the verb comes next:

• I dag kan hun ...

Structure:

• i dag = first element
• kan = second element, so it must come here
• hun = subject after the verb

This is very common in Danish:

• I går kunne hun ikke sove.
• Nu forstår jeg det.
• Måske kommer han senere.

Because næsten ikke is the natural way to say almost not / hardly in Danish.

So:

• Hun kunne næsten ikke bevæge håndleddet = she could hardly move her wrist

If you said ikke næsten, it would sound wrong in this context.

A useful way to understand it is:

• næsten ikke = not quite able to, barely able to
• English often translates this more naturally as hardly

Because after a modal verb like kan or kunne, Danish uses the bare infinitive.

So you get:

• kan skrive
• kunne bevæge

not:

• kan at skrive
• kunne at bevæge

This works like English:

• She can write
• She could move

not:

• She can to write
• She could to move

Håndleddet is the definite form of håndled.

• et håndled = a wrist
• håndleddet = the wrist

Here, the definite form is used because it refers to a specific wrist that the listener can identify from the context: her wrist.

With body parts, Danish often uses the definite form where English often uses a possessive:

• Hun løftede hånden. = She raised her hand.
• Han brækkede benet. = He broke his leg.
• Hun kunne ikke bevæge håndleddet. = She couldn’t move her wrist.

It could, but it is usually less natural unless you want extra emphasis or contrast.

• håndleddet simply means the wrist, with the owner understood from context
• hendes håndled would stress that it is her wrist, perhaps to distinguish it from someone else’s

So in an ordinary sentence, håndleddet is the more natural choice.

Bevæge means to move.

In this sentence:

• bevæge håndleddet = move the wrist

It refers to physical movement. So the idea is that yesterday she was barely able to make her wrist move.

You may also see bevæge sig, which means to move oneself / to move around:

• Hun bevæger sig langsomt. = She moves slowly.

But here it is not reflexive, because the wrist is the direct object:

• bevæge håndleddet

It means a short message.

• en besked = a message
• kort = short

In context, it could mean something like:

• a short text message
• a short written note
• a brief message in general

It does not have to mean a long formal piece of writing. The point is that today she is able to write at least a small amount.

Here it is about ability in the past, not politeness.

Because the sentence includes i går = yesterday, kunne clearly means:

• she was able to
• she could

It does not mean something like a polite request, as in English Could you help me?

So:

• Hun kunne næsten ikke bevæge håndleddet i går = Yesterday she was hardly able to move her wrist.

Because men joins two separate main clauses, and each clause needs its own subject.

The sentence is:

• Hun kunne næsten ikke bevæge håndleddet i går
• men i dag kan hun skrive en kort besked

The second clause is complete on its own, so it needs hun.

This is the same as English:

• She could hardly move her wrist yesterday, but today she can write a short message.

Yes. Danish is fairly flexible, but the position affects word order because of the V2 rule.

For example, these are possible:

• Hun kunne næsten ikke bevæge håndleddet i går.
• I går kunne hun næsten ikke bevæge håndleddet.

And:

• I dag kan hun skrive en kort besked.
• Hun kan skrive en kort besked i dag.

Both versions are grammatical, but when you move i går or i dag to the front, the verb must come right after it.