Learn Hardcore Dutch: De docent vroeg een leerling om het bord schoon te maken. - The teacher asked a pupil to clean the board.

In Dutch, when you ask someone to do something, you usually follow vragen with an infinitive clause introduced by om + te. The pattern is:
iemand vragen om iets te doen
So om marks the start of that infinitive clause. Without om, it’s either ungrammatical or sounds very clipped.

Why is the separable verb schoonmaken split into schoon te maken?

Schoonmaken is a separable verb (particle + verb). In an infinitive construction with te, Dutch splits the verb:
particle (schoon) + te + main verb (maken).
That’s why you see schoon te maken instead of one word.

What is the function of het bord in the sentence?

Het bord (“the board”) is the direct object of the verb schoonmaken inside the infinitive clause. It answers the question What is being cleaned.

Why is there een leerling (an indefinite “student”) rather than de leerling (the specific “student”)?

Using een indicates “any student” or “one of the students.” If you said de leerling, you’d refer to a particular student already known to both speaker and listener.

Why is the verb vroeg (asked) in the simple past tense and not in the perfect tense?

Dutch often uses the simple past to report past actions, especially in narratives or plain statements. You could also use the perfect to focus on completion, but the simple past (vroeg) is perfectly natural here.

Can I replace vroeg with heeft gevraagd and keep the same meaning?

Yes. You can say:
De docent heeft een leerling gevraagd om het bord schoon te maken.
The basic meaning stays the same; you’re just using the present perfect instead of the simple past.

Why isn’t there a dat (that) introducing the clause, like in English “that …”?

Because this is an infinitive clause (om + te), not a subordinate clause with its own conjugated verb. Infinitive clauses never take dat.

Could I use laten (to let/have someone do something) instead of vragen here?

Yes, but the nuance changes.
De docent liet een leerling het bord schoonmaken.
Here liet (from laten) means the teacher caused or instructed the student to do it, rather than politely asking.