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Question

"A esperar que termines de discutir con tu esposa?

I'm just wondering what "que" means in this sentence.

And also, why does it start with "a"?

Answer 1

I'm just wondering what "que" means in this sentence.

In English, ¨that¨is many times left out of the sentence, it is implied.
In Spanish, ¨that¨is always translated with ¨que¨.

Example:
Do you expect me to read the book?
¿Espera que lea el libro? ("lea" is a subjunctive)