Korean · Pronunciation Guide
mun-je. A rounded first vowel, a forward n, and a ㅈ that voices after the nasal onto the pure ㅔ of 'bet'.
munje
Two beats. Every consonant is voiced and the whole word runs without a break, which is why it sounds so smooth despite containing three consonants in five sounds.
Listen: 문제
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우리에게 문제가 생겼어요.
Uriege munjega saenggyeosseoyo.
We have a problem.
Listen: 우리에게 문제가 생겼어요.
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우리에게 문제가 생겼어요 attaches the subject particle to a vowel-final noun and finishes with 생겼어요, where the ㅆ resyllabifies into a double s. The whole sentence is voiced right up until that point.
Putting breath on the ㅈ into 'mun-che' is the usual error with the plain series after a nasal, when it should be fully voiced. The other issue is the ㅔ, which English speakers turn into 'jay' by adding a glide, whereas Korean holds one position through the vowel.