엄마가 너 언제 오냐고 물어봤어

The line, explained

Every family drama has this phone call. One sibling is at home, the other is far away and busy, and the one at home says, very casually, 'Mom asked when you're coming.' It is not really a message. It is a nudge, delivered through someone else's mouth. Korean makes this easy with -냐고, the ending that turns a question into reported speech. Once you have it, you can pass along anything anyone has ever asked you.

엄마가 is 'Mom' with the subject marker 가, so Mom is the one doing the asking. 엄마 is the warm, everyday word, the one adults use too; 어머니 is more formal. 너 is the casual 'you', the sibling on the other end of the line. Korean puts the person the question is about right up front, before the question word, so 엄마가 너 sets the scene: Mom, about you, something.

언제 means when. 오다 is to come, and Mom's original question was probably 언제 와?, 'when are you coming?' To report it, the question is put into a plain form ending in -냐: 언제 오냐. Then -고 wraps it up as a quote, giving 언제 오냐고, '(asking) when you're coming'. The question word 언제 stays inside, exactly where it was. The same trick works with yes-or-no questions: 오냐고 alone means 'whether you're coming'.

물어봤어 is the casual past of 물어보다, to ask. It is built from 묻다, to ask, and 보다, to try, which softens it into 'asked, checked with'. You could also say 물었어 or simply 했어, and the meaning stays the same. Notice the order: who asked, about whom, the question itself with -냐고, and then the verb of asking at the very end. This is how Korean reports any question: the full question first, then the reporting verb.

Read it like a sibling who is trying to sound casual and failing: eom-ma-ga neo eon-je o-nya-go, then a slight pause, mu-reo-bwa-sseo. Then try it with other questions Mom might ask: 밥 먹었냐고, whether you have eaten; 잘 지내냐고, whether you are doing okay; 왜 연락 안 하냐고, why you never call. Each one is a tiny drama of its own, and -냐고 is how all of them get delivered.