If that G is just a brief passing note on top of the B minor chord, then it doesn't need a name - It's just B minor. Most of the time a G note would just appear and disappear quickly - it wouldn't stay long enough to make a name for itself.
If you stay on it long enough then it would have the name we said Bm(b6) - but the reason we rarely see it is because the G is an avoid tone as it makes a very dissonant minor second interval with the F#. Usually the quicker we pass through it, the better.
Anyway, it's NOT an extended chord, as it has no 7th, so it's an added chord
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