Votre demande est déjà enregistrée au secrétariat.

In this sentence, demande means request or application, depending on the context.

For example:

• in an office or administration, it might mean an application
• in a more general situation, it might mean a request

So Votre demande is your request / your application.

Votre means your.

It is used:

• when speaking to one person formally
• or to more than one person

French uses votre instead of ton/ta/tes when the situation is polite, official, or formal. Since this sentence sounds administrative, votre is the natural choice.

Because this sentence uses être + past participle, which gives a passive meaning.

So:

• enregistrer = to record / to register
• est enregistrée = is registered / has been recorded

The idea is that the request has already been entered into the system or officially recorded.

This is similar to English:

• The request is registered
• The request has been recorded

Because demande is a feminine singular noun, and the past participle agrees with it here.

• demande = feminine singular
• so the participle becomes enregistrée

Compare:

• Le dossier est enregistré → masculine singular
• La demande est enregistrée → feminine singular
• Les demandes sont enregistrées → feminine plural

Déjà means already.

It comes between est and enregistrée:

• est déjà enregistrée

That placement is very natural in French. It modifies the whole idea of the request being registered.

So the sentence means that the action is already completed:

• Your request is already registered
• Your request has already been recorded

Au secrétariat means at the office, with the administrative office, or in the secretariat.

Literally:

• à = at / to
• le secrétariat = the secretariat
• au = à + le

In practice, secrétariat often refers to the administrative office or front office of an organization, not just to one secretary.

So au secrétariat suggests that the request has been officially recorded by or in the administrative office.

Because in French, à + le contracts to au.

So:

• à + le = au
• à + les = aux

That is why you say:

• au secrétariat not
• à le secrétariat

Sometimes yes, but the everyday meaning in French is often broader and more practical.

In this kind of sentence, le secrétariat usually means:

• the administrative office
• the office staff
• the department handling paperwork

So a learner should not always think only of the English institutional word secretariat. In many contexts, office or administrative office is a more natural translation.

It could, but it would sound a little different.

• est déjà enregistrée focuses on the current result/state: the request is now on file.
• a déjà été enregistrée focuses more clearly on the completed action: it has already been recorded.

In many real contexts, est déjà enregistrée is very natural because the important point is the present status of the request.

Yes, it sounds fairly formal and administrative.

Clues:

• Votre instead of an informal ta
• demande in an official sense
• enregistrée as administrative vocabulary
• au secrétariat referring to an office or department

This is exactly the kind of sentence you might see in:

• an email from an office
• a school or university administration
• a hospital or clinic
• a government service
• a company handling applications

Yes. Enregistrer has several related meanings, including:

• to record audio/video/data
• to register or enter something officially
• to save information

In this sentence, it means something like:

• to register
• to log
• to officially record

So the request has been entered into the records/system.

The word order is quite standard for French:

• Votre demande = subject
• est = verb
• déjà enregistrée = predicate/passive result
• au secrétariat = place or administrative location

French often places short adverbs like déjà after the conjugated verb:

• est déjà enregistrée
• a déjà commencé
• sont déjà prêts

So the position of déjà is something worth noticing and remembering.