A biztosító elutasítja a kárigényt, holott van bizonyíték.

Because the sentence has a definite direct object: a kárigényt = the claim.

Hungarian verbs often have two conjugation patterns:

• indefinite conjugation: used when there is no definite object
• definite conjugation: used when the object is definite

Here:

• elutasít = he/she/it rejects (indefinite)
• elutasítja = he/she/it rejects it / the ... (definite)

Since a kárigényt is specific and definite, Hungarian uses elutasítja.

The -t marks the accusative case, which usually shows the direct object of the verb.

So:

• kárigény = claim for damages / insurance claim
• kárigényt = the claim as the object of rejects

This is very common in Hungarian:

• kérelem = request
• kérelmet = request (as object)
• könyv = book
• könyvet = book (as object)

Because the insurer rejects the claim, claim has to be in the accusative.

Kárigény is a compound noun:

• kár = damage, loss
• igény = claim, demand, entitlement

Together, kárigény means something like:

• claim for damages
• insurance claim
• compensation claim

In this sentence, the most natural English translation is usually insurance claim or damage claim, depending on context.

Here, biztosító means insurer or insurance company.

It comes from the idea of insuring. Be careful not to confuse related words:

• biztosító = insurer / insurance company
• biztosítás = insurance
• biztosít = to insure / to secure / to ensure, depending on context

So A biztosító means The insurer.

Hungarian uses definite articles a and az much like English the.

Here:

• A biztosító = The insurer
• a kárigényt = the claim

Hungarian chooses:

• a before most consonant sounds
• az before vowel sounds

Examples:

• a biztosító
• a kárigény
• az autó
• az ember

So the article is there because both nouns are understood as specific.

Holott means something like:

• although
• even though
• whereas
• despite the fact that

In this sentence, it introduces a contrast:

• A biztosító elutasítja a kárigényt, holott van bizonyíték.
• The insurer rejects the claim, even though there is evidence.

It often sounds a little formal or written in style.

Yes, but the nuance changes a bit.

• bár = although
• pedig = often while / whereas / and yet
• holott = even though / whereas, often with a slightly more formal or argumentative tone

For example:

• Bár van bizonyíték, a biztosító elutasítja a kárigényt.
  = Although there is evidence, the insurer rejects the claim.

• Van bizonyíték, pedig a biztosító elutasítja a kárigényt.
  This structure is possible, but it feels a bit different in emphasis.

• Holott van bizonyíték...
  sounds like despite the fact that there is evidence...

So holott is a good choice if the speaker wants to stress the contradiction.

Because bizonyíték here is being used in an indefinite existential sense: there is evidence.

Hungarian often leaves out the article in this kind of statement:

• Van idő. = There is time.
• Van pénz. = There is money.
• Van bizonyíték. = There is evidence.

If you said van a bizonyíték, that would usually sound odd unless you mean something very specific like the evidence is there in a special contrastive context.

So van bizonyíték is the normal form.

Van is the 3rd person singular form of lenni = to be.

Here it means there is:

• van bizonyíték = there is evidence

Hungarian often uses van for existence or availability.

Compare:

• Bizonyíték van. = There is evidence.
  This version gives more emphasis to the fact that evidence exists.

• Van bizonyíték. = There is evidence.
  This is the more neutral wording.

Yes, it is in the present tense:

• elutasítja = rejects
• van = there is

Hungarian often uses the present tense in the same situations English does:

• for current events
• for general statements
• for vivid or immediate description

So the sentence can mean:

• The insurer rejects the claim, even though there is evidence.

Depending on context, English might also choose a different tense, but the Hungarian present is completely normal here.

Yes. Hungarian word order is flexible, and changing it often changes emphasis rather than basic meaning.

The original sentence is fairly neutral:

• A biztosító elutasítja a kárigényt, holott van bizonyíték.

Other possible versions:

• Holott van bizonyíték, a biztosító elutasítja a kárigényt.
  This puts the contrast clause first: Even though there is evidence...

• A biztosító a kárigényt utasítja el, holott van bizonyíték.
  This can put more focus on the claim as the thing being rejected.

• A kárigényt a biztosító elutasítja, holott van bizonyíték.
  This foregrounds the claim.

So yes, word order can move around, but it usually affects information structure and emphasis.

Because Hungarian often builds object information into the verb itself.

In elutasítja, the -ja ending already shows that the verb is being used with a definite object. So the verb form itself helps signal something like rejects it / rejects the specific thing.

That does not mean the verb alone always fully replaces the object, but it is one reason Hungarian does not need to use pronouns in the same way English does.

Hungarian often leaves pronouns out when they are already clear from the verb or context.

A fairly literal breakdown is:

• A biztosító = the insurer
• elutasítja = rejects it / rejects
• a kárigényt = the claim
• holott = even though / although / whereas
• van = there is
• bizonyíték = evidence

So, very literally:

The insurer rejects the claim, even though there is evidence.

Or almost word-for-word:

The insurer rejects the damage-claim, although there is evidence.