Forhandleren tjekker serienummeret med henblik på at udbedre manglen.

Because Danish usually adds the definite article to the end of the noun.

• en forhandler = a dealer / a retailer
• forhandleren = the dealer / the retailer

So -en here is basically the Danish equivalent of the.

That is because Danish nouns have grammatical gender.

• en forhandler → forhandleren
• en mangel → manglen
• et serienummer → serienummeret

So:

• nouns that take en usually form the definite with -en
• nouns that take et usually form the definite with -et

Tjekker is the present tense of at tjekke.

• at tjekke = to check
• tjekker = checks / is checking

A useful Danish rule is that the present tense is often formed with -r. Also, unlike English, Danish does not change the verb for different persons:

• jeg tjekker
• du tjekker
• han tjekker
• de tjekker

The form stays the same.

Yes. Tjekke is a normal modern Danish verb, borrowed from English check but adapted to Danish spelling and grammar.

So even though it looks familiar, you should treat it like an ordinary Danish verb:

• at tjekke
• tjekker
• tjekkede
• har tjekket

It is a fixed expression meaning something like:

• with a view to
• for the purpose of
• with the aim of
• in order to

It is more formal than everyday for at. You often see it in official, legal, or business-style Danish.

Because the full expression is med henblik på at + infinitive.

• med henblik på is the fixed phrase
• at udbedre is the infinitive, meaning to remedy / to fix

So you should learn med henblik på at as a chunk when it is followed by a verb.

Here at is the infinitive marker, similar to English to.

• at udbedre = to remedy / to fix
• at tjekke = to check

Be careful: Danish at is not always translated by English to, but in this kind of structure, that is usually the easiest way to understand it.

Udbedre is a fairly formal verb meaning to remedy, to rectify, or to repair a problem, defect, or damage.

It is often used for things like:

• faults
• defects
• damage
• deficiencies

It sounds a bit more formal and technical than a very everyday verb like fikse.

Because the sentence refers to a specific defect or deficiency, not just any defect.

• en mangel = a defect / a deficiency
• manglen = the defect / the deficiency

So the definite form shows that this is a particular known problem.

Yes. That would be more everyday and more direct.

Compare:

• for at udbedre manglen = simpler, more natural in everyday speech
• med henblik på at udbedre manglen = more formal, more bureaucratic, more written-style

Both are correct, but they give a slightly different tone.

The sentence follows normal Danish main-clause word order:

• Forhandleren = subject
• tjekker = verb
• serienummeret = object
• med henblik på at udbedre manglen = purpose phrase

So the structure is basically:

Subject + verb + object + purpose expression

That is very similar to standard English word order.