Arra hivatkozva, hogy érvényes az árajánlat, a teljes kárösszeget kérjük.

It means something like by referring to that, on the grounds that, or citing the fact that.

Here:

• hivatkozni valamire = to refer to something / to cite something
• hivatkozva is the -va/-ve form, which often works like while doing X, by doing X, or having done X

So Arra hivatkozva, hogy... is a very common formal pattern meaning:

• by referring to the fact that...
• on the basis that...
• claiming that...

It is especially common in administrative, legal, insurance, or complaint-style language.

Arra is the -ra/-re form of az and here it points forward to the hogy clause.

So:

• arra = to that / to the fact that
• hogy érvényes az árajánlat = that the quote/estimate is valid

Together:

• arra hivatkozva, hogy... = referring to the fact that...

This forward-pointing structure is very common in Hungarian:

• arra gondolok, hogy... = I’m thinking that...
• arra emlékszem, hogy... = I remember that...
• arra hivatkozva, hogy... = citing the fact that...

So arra is not random: it is required by the verb pattern hivatkozik valamire.

The verb is hivatkozik, and it normally takes a complement with -ra/-re:

• hivatkozik valamire = refers to something / cites something

Examples:

• A szerződésre hivatkozik. = He/She is referring to the contract.
• A törvényre hivatkoznak. = They are citing the law.

In your sentence, instead of a noun like a szerződésre, Hungarian uses:

• arra ... hogy... = to the fact that...

So the structure is fully regular:

• hivatkozik + -ra/-re
• here: arra hivatkozva

Hivatkozva is the adverbial participle form of hivatkozik.

The ending -va/-ve often expresses:

• while doing
• by doing
• having done
• sometimes a looser idea like with reference to

So Arra hivatkozva, hogy... is not a literal everyday while referring to that, but a set formal expression meaning:

• citing that...
• on the basis that...
• arguing that...

In this sentence, it introduces the reason or justification for the request.

Both are possible, but the word order changes the emphasis slightly.

• Az árajánlat érvényes. = The quote is valid.
  This is the more neutral order.
• Érvényes az árajánlat. = The quote is valid / It is valid, the quote.
  This puts a little more focus on érvényes.

In your sentence, the speaker is emphasizing the claim of validity, because that is the basis for requesting the full amount. So érvényes comes first naturally.

Hungarian word order is strongly connected to information structure, not just fixed subject-verb-object order.

Árajánlat usually means price quote, quotation, or estimate.

It is a compound:

• ár = price
• ajánlat = offer / proposal

Depending on context, it could mean:

• a business quote
• a repair estimate
• a service quotation

In an insurance or damage context, it often refers to a repair estimate or cost quotation.

Because Hungarian uses the definite article when referring to a specific, identifiable thing.

• az árajánlat = the quote / the estimate

Here, it is clearly a particular quote already known to both sides, not just any quote. That is why the definite article is used.

Also, az is used instead of a because the next word begins with a vowel:

• a szerződés
• az árajánlat

It means the full amount of damages or the entire damage amount.

Breakdown:

• a = the
• teljes = full / complete
• kárösszeg = amount of damage / compensation amount
• kárösszeget = accusative form, because it is the object of the verb

Kárösszeg is also a compound:

• kár = damage / loss
• összeg = sum / amount

So literally it is something like the full damage amount.

Because it is in the accusative case, which marks the direct object.

The verb kérjük means we ask for / we request, and the thing being requested is:

• a teljes kárösszeget

So:

• kárösszeg = base form
• kárösszeget = direct object form

This -t is one of the most important Hungarian case endings.

Because Hungarian verbs have definite and indefinite conjugation.

Here the object is definite:

• a teljes kárösszeget = the full damage amount

So Hungarian uses the definite conjugation:

• kérünk = we ask (for something indefinite / unspecified)
• kérjük = we ask for it / we request it with a definite object

That is why the sentence has:

• a teljes kárösszeget kérjük

This is a very important grammar point in Hungarian.

No, not in this sentence.

Kérjük can sometimes function like please, especially in public notices:

• Kérjük, várjanak! = Please wait!

But here it is a normal verb with a definite object:

• a teljes kárösszeget kérjük = we request the full amount of damages

So in this sentence it means we request / we ask for, not simply please.

Because Hungarian usually leaves out subject pronouns when the verb ending already shows the subject.

• kérjük already tells you the subject is we

So adding mi would usually be unnecessary unless you want contrast or emphasis.

For example:

• Mi kérjük a teljes kárösszeget. = We are the ones requesting the full amount.

Without emphasis, Hungarian normally just says:

• A teljes kárösszeget kérjük.

Yes, very formal.

The whole style sounds like administrative, legal, insurance, or complaint language. In more everyday Hungarian, people might choose a simpler structure.

For example, a less formal version could be:

• Mivel az árajánlat érvényes, a teljes kárösszeget kérjük.
• Azért kérjük a teljes kárösszeget, mert az árajánlat érvényes.

But Arra hivatkozva, hogy... sounds especially formal and argument-based, as if giving justification in writing.

The commas separate the introductory phrase from the main clause.

Structure:

• Arra hivatkozva, hogy érvényes az árajánlat, = introductory justificatory clause
• a teljes kárösszeget kérjük. = main clause

The first comma appears before the hogy clause because this is the standard punctuation in that expression.

The second comma closes the introductory unit before the main statement begins.

So the commas help show that the first part gives the reason or basis, and the second part gives the actual request.

Not very naturally.

A very literal version would be something like:

• Referring to that, that the quote is valid, we request the full damage amount.

That is grammatical only in a very rough, word-by-word sense. Natural English would usually reshape it, for example:

• Since the quote is valid, we request the full amount of damages.
• On the grounds that the estimate is valid, we request the full amount of compensation.
• As the estimate is valid, we are requesting the full damage amount.

This is a good example of why Hungarian structures often should not be translated too literally.