Responses
It was mentioned that females of the Hayase family are the ones tasked to watch Fushi, the mother will try until they have a girl to pass on the job. I don't remember them saying that they kill the male children so maybe they became a branch family or they're abandoned/exiled. I think it was just Kahak who was the only child, since the mother(former chosen) died before there was a girl and apparently there wasn't anyone else suitable/available? That's what I think.

I thought kahak had no kids...... Who's the woman that bore his child.. ? ..... My yaoi ass was rooting for him, lol. Mizuha is just like hayase, so I don't really like her. Kahak is better.....