"어머니는 눈치가 있게 물어봤어요."
Translation:The mother asked tactfully.
눈치가 있게, or 눈치있게 can be used in front of someones acts. the trandlation it correlates to in here is tactfully, although i would use carefully. 그 사람은 눈치가 있게 자리를 떠납니다., for example, can be translated as the man moves out of a seat tactfully., or the man moves out of the place carefully.