No quiero irme hasta que mi hija vea otra vez a la yegua y al pato.

Because irse and ir are related but not identical.

• ir = to go
• irse = to go away / to leave

So:

• No quiero ir = I don’t want to go
• No quiero irme = I don’t want to leave / go away

In this sentence, irme gives the idea of leaving the place. The me is part of the pronominal verb irse.

Also, after querer, Spanish often uses the infinitive directly:

• Quiero irme
• No quiero irme

So irme is the infinitive irse, with the pronoun attached to the end.

Because after hasta que, Spanish often uses the subjunctive when the action has not happened yet.

Here, the daughter has not seen the mare and the duck again yet. The speaker is talking about a future point:

• I don’t want to leave until my daughter sees them again.

Since that seeing is still pending, Spanish uses:

• hasta que mi hija vea...

not:

• hasta que mi hija ve...

So vea is the present subjunctive of ver.

Hasta que can be followed by either indicative or subjunctive, depending on whether the action is seen as real/already habitual, or still future/expected.

In this sentence, the action is still in the future from the speaker’s point of view:

• first, the daughter sees the animals again
• then, the speaker leaves

That is why Spanish uses the subjunctive:

• No quiero irme hasta que mi hija vea...

Compare:

• Me quedé allí hasta que mi hija vio a la yegua.
  = The event actually happened in the past, so indicative.

A useful shortcut:

• future or not yet completed after hasta que → usually subjunctive
• completed/past fact → indicative

Vea is the first person singular and third person singular present subjunctive form of ver.

Present subjunctive of ver:

• yo vea
• tú veas
• él/ella/usted vea
• nosotros veamos
• vosotros veáis
• ellos/ellas/ustedes vean

In your sentence, the subject is mi hija, which is third person singular, so:

• mi hija vea

This is the personal a.

Spanish often uses a before a specific direct object when it refers to:

• a person
• or sometimes an animal, especially when it is individualised, familiar, or treated as a distinct being

Here, la yegua and el pato are specific animals, so Spanish uses:

• a la yegua
• al pato

This is very natural if the animals are known or important in the context.

Compare:

• Veo un pato en el lago.
  = I see a duck in the lake.
  (more generic; no personal a)

• Veo al pato todos los días.
  = I see the duck every day.
  (a specific duck)

So the a here does not mean to. It marks the direct object.

Because a + el contracts to al.

So:

• a + el pato → al pato

This contraction is mandatory in normal Spanish.

Compare:

• a la yegua → no contraction, because a + la does not contract
• a el pato → becomes al pato

The same thing happens with de + el:

• del = de + el

Otra vez means again.

In this sentence, it modifies vea:

• vea otra vez = see again

The idea is that the daughter has seen the mare and the duck before, and the speaker wants that to happen one more time.

Its placement is flexible, but this position is very natural:

• hasta que mi hija vea otra vez a la yegua y al pato

You may also hear:

• hasta que mi hija vuelva a ver a la yegua y al pato

That is another common way to say see again in Spanish:

• volver a + infinitive = to do something again

So both are possible:

• vea otra vez
• vuelva a ver

Because the definite articles la and el suggest that these are specific, known animals.

• la yegua = the mare
• el pato = the duck

This sounds like both speaker and listener already know which animals are being talked about.

If you said:

• a una yegua y a un pato

that would mean a mare and a duck, with a less specific or more indefinite meaning.

So the sentence as written implies particular animals, not just any mare and any duck.

Not here.

You use hasta que when what follows is a clause with a conjugated verb:

• hasta que mi hija vea...

Because vea is a full verb form, que is needed.

You use just hasta before:

• a noun
• a pronoun
• an infinitive in some contexts
• a time expression

Examples:

• hasta mañana
• hasta el final
• hasta verlo (less parallel to your sentence, but possible in some structures)

So in your sentence, hasta que is the correct structure because mi hija vea... is a clause.

The original word order is very natural:

• No quiero irme hasta que mi hija vea otra vez a la yegua y al pato.

But Spanish allows some variation.

For example:

• No quiero irme hasta que mi hija vea a la yegua y al pato otra vez.

This is also understandable, though vea otra vez a la yegua y al pato often sounds slightly smoother.

You could also emphasize the subject:

• No quiero irme hasta que sea mi hija quien vea otra vez a la yegua y al pato.

But that changes the focus.

So the original sentence is a good, standard way to say it.

Yes. Both are correct.

• No quiero irme
• No me quiero ir

Both mean I don’t want to leave.

This happens because with a verb like querer followed by an infinitive, the pronoun can usually go:

• attached to the infinitive: irme
• or before the conjugated verb: me quiero ir

So both are standard and natural.

Very roughly:

• No quiero irme may sound a little more neutral or compact.
• No me quiero ir may feel slightly more conversational in some contexts.

But the difference is small, and both are common in Spain.