Learn Hardcore German: Meine Freundin hat den Wunsch, die Frist zu verlängern; darüber sprechen wir mit der Chefin. - My friend female has the wish to extend the deadline; we will talk about that with the boss.

Because Wunsch is masculine and here it’s the direct object of hat (to have). In German, masculine nouns take den in the accusative singular. Compare:

Nominative: Der Wunsch ist verständlich. (subject)
Accusative: Sie hat den Wunsch. (direct object)

Could I say einen Wunsch instead of den Wunsch?

You’ll hear both, but den Wunsch, … is the more idiomatic, set phrase when you specify the content of the wish with an infinitive clause. Einen Wunsch emphasizes the existence of a wish in general. Natural alternatives are:

Meine Freundin möchte die Frist verlängern. (simpler, very common)
Meine Freundin wünscht sich eine Fristverlängerung. / … wünscht sich, dass die Frist verlängert wird. (more explicit)

Why is there a comma before die Frist zu verlängern?

German requires a comma before a zu-infinitive group when it’s linked to a “head word” like a noun or pronoun that announces it—in this case den Wunsch. So the comma is mandatory here. Without such a head word, the comma can often be optional.

What is the role of zu verlängern here?

It’s a zu + infinitive clause that specifies the content of the wish. The understood subject of zu verlängern is the same as the main clause subject (Meine Freundin), unless you rephrase to make a different subject clear (see below).

Should it be verlängern zu lassen if the boss is the one who extends the deadline?

Use verlängern lassen if you want to stress that someone else performs the action:

… den Wunsch, die Frist verlängern zu lassen.
In everyday speech, people often still say die Frist zu verlängern when they mean “to get the deadline extended,” and context does the rest. If you need full clarity, use a dass-clause:
… den Wunsch, dass die Chefin die Frist verlängert.

Why is there a semicolon; could I use a comma or a period instead?

You’re linking two independent main clauses. German allows:

Semicolon to signal a close but clear separation (as here).
Period to separate them completely.
A comma between independent clauses without a conjunction is allowed in modern usage when the clauses are closely related, but many writers prefer a semicolon or period for clarity.

Why is it darüber sprechen wir and not wir sprechen darüber?

German is a verb-second (V2) language. When you front darüber for emphasis or topicalization, the finite verb (sprechen) must still be in second position, so the subject (wir) comes after the verb: Darüber sprechen wir …
The neutral order Wir sprechen darüber … is also perfectly correct.

What exactly does darüber refer to, and could I say über das or dazu?

Darüber is a pronominal adverb meaning “about that,” referring to the previously mentioned idea “extending the deadline.”

Über das usually needs an explicit noun: über das Thema.
Dazu uses the preposition zu, which doesn’t fit with sprechen here. With sprechen, the idiomatic choice is über + Akkusativ → darüber.

Can I move mit der Chefin around in the sentence?

Yes. Word order is flexible for emphasis:

Wir sprechen mit der Chefin darüber. (very natural)
Wir sprechen darüber mit der Chefin.
Mit der Chefin sprechen wir darüber. (emphasis on who you’ll talk to) All are correct.

Why is it mit der Chefin (der, not die)?

The preposition mit always takes the dative case. Chefin is feminine; dative feminine singular is der. So: mit der Chefin. (Masculine would be mit dem Chef.)

What’s the nuance of Chefin vs. Chef?

Chefin is the feminine form (“female boss”). Chef is masculine and is also often used generically (“boss” with unknown gender). Gender-neutral options include die Vorgesetzte / der Vorgesetzte (manager/superior) or die Leitung (management/leadership) depending on context.

Does Freundin always mean “girlfriend,” or can it be just “female friend”?

It’s ambiguous. Meine Freundin commonly implies a romantic partner, but context can also make it mean “a (female) friend.” If you want to avoid ambiguity:

romantic partner: meine feste Freundin / meine Partnerin
just a friend: eine Freundin von mir

Why die Frist—what is a Frist, and how is it different from Termin?

Frist (feminine: die Frist) is a time limit or deadline (a period by which something must be done). Termin is an appointment or a specific date/time. Typical collocations:

eine Frist setzen / verlängern / einhalten / versäumen
einen Termin vereinbaren / verschieben / absagen

With sprechen, is it always über + accusative?

Yes. You sprechen über etwas (accusative): über wen? über was? That’s why the pronominal adverb is darüber. Synonyms behave similarly: reden/sprechen/ sich unterhalten über + Akkusativ.

Does sprechen wir refer to the future even though it’s present tense?

Yes. German often uses the present tense for near-future plans: Darüber sprechen wir morgen. You can also say werden sprechen for emphasis on futurity, but it’s not required here.

Could I avoid the zu-infinitive and use a dass-clause?

Absolutely. Options include:

… hat den Wunsch, dass die Frist verlängert wird.
If you want to name the agent: … dass die Chefin die Frist verlängert. These make the subject of the extending action explicit.