Nous regardons la vidéo enregistrée pour entendre notre prononciation.

In French, regarder means “to watch / to look at” and it takes a direct object, with no preposition:

• regarder quelque chose = to look at / watch something
• Nous regardons la vidéo. = We are watching the video.

Using regarder à in this context would be wrong.
(Regarder à exists in some fixed expressions but with different meanings, e.g. regarder à la dépense = to be careful with spending.)

French normally uses the simple present for both:

• “We watch”
• “We are watching”

So Nous regardons la vidéo can mean either “We watch the video” (habit) or “We are watching the video” (right now).
Context tells you which meaning is intended. There is a progressive form (nous sommes en train de regarder), but it’s only used when you really want to emphasize “in the middle of doing it right now.”

• la vidéo = the video (a specific one, already known in the context)
• une vidéo = a video (not specified which one)

In the sentence, the group is clearly referring to a particular video they recorded, so French naturally uses the definite article: la.

Also, vidéo is feminine in French, so it takes la, not le.

For many nouns in French, gender is arbitrary and must simply be learned with the word.

• la vidéo → feminine
• une vidéo → a video
• les vidéos → the videos

There is no logical rule that forces vidéo to be feminine; it’s just how the word has evolved in French. You have to memorize it as la vidéo.

Enregistrée is the past participle of enregistrer (“to record”), used here as an adjective:

• enregistrer → past participle enregistré (masc.) / enregistrée (fem.)

It describes la vidéo, which is feminine singular, so it must agree:

• masculine singular: un son enregistré
• feminine singular: une vidéo enregistrée
• feminine plural: des vidéos enregistrées

In French, past participles and many adjectives that describe a noun usually go after that noun:

• la vidéo enregistrée = the recorded video

They mean almost the same thing in this sentence:

• la vidéo enregistrée
• la vidéo qui a été enregistrée = the video that was recorded

The version with qui a été is a relative clause and sounds a bit more explicit or heavy.
The shorter la vidéo enregistrée is more natural in everyday speech and writing, unless you need to add more information (e.g. la vidéo enregistrée hier).

Yes. pour + infinitive is the standard way to express purpose (“in order to …”):

• pour entendre = in order to hear / to hear
• Nous regardons la vidéo enregistrée pour entendre…
  = We watch the recorded video in order to hear …

You could also say afin d’entendre, which is a bit more formal, but pour + infinitive is the most common everyday structure for expressing goal/purpose.

Both relate to hearing, but there’s a nuance:

• entendre = to hear (the sound reaches your ears, more passive)
• écouter = to listen (to) (you actively pay attention)

In this context, both are actually possible:

• pour entendre notre prononciation
• pour écouter notre prononciation

Many speakers would probably prefer écouter here, because the learners are actively listening to their own pronunciation.
Using entendre slightly emphasizes the idea of hearing how it sounds, but it’s not wrong.

Notre is used because:

• prononciation is singular here: it refers to the way we pronounce as a group, our collective pronunciation.
• notre = our (for one thing we own/share)
• nos = our (for more than one thing)

So:

• notre prononciation = our pronunciation (considered as one system/style)
• nos prononciations would suggest multiple distinct pronunciations, which is possible in some contexts but not the natural default here.

In French, possessive adjectives like mon, ton, son, notre, votre, leur replace the definite article; you don’t use both:

• notre prononciation = our pronunciation
• not: la notre prononciation (incorrect)

So you say:

• la prononciation = the pronunciation
• notre prononciation = our pronunciation

You never add another article in front of notre.

The direct object pronoun for la vidéo is la.
In standard French word order, object pronouns go before the conjugated verb:

• Nous la regardons pour entendre notre prononciation.
  = We watch it to hear our pronunciation.

You cannot say Nous regardons la pour…; that would be wrong in French.

One careful pronunciation (in IPA) is:

• Nous regardons la vidéo enregistrée pour entendre notre prononciation.
  → /nu ʁə.ɡaʁ.dɔ̃ la vi.de.o ɑ̃.ʁə.ʒis.tʁe puʁ ɑ̃.tɑ̃dʁ nɔtʁ pʁɔ.nɔ̃.sja.sjɔ̃/

Key points for learners:

• Final -s in nous is silent: /nu/
• Final -ons in regardons is pronounced /ɔ̃/ (nasal vowel), the -s is silent.
• vidéo has 3 syllables: vi-dé-o.
• enregistrée: nasal en- /ɑ̃/, and final -ée is /e/ (not /ɛ/).
• entendre: nasal en- again, final -re is just /ʁ/; the final e is silent.
• prononciation: the -tion ends with /sjɔ̃/ (nasal vowel).

There are no obligatory liaisons here that you must make; you can pronounce each word separately and still sound natural.