Can I give my zakat to a poor woman with children who is unable to support her family?

You may help this woman with zakat if she is poor. You say that she does not have a regular income. It may be that her income, though irregular, is sufficient to meet her needs and, in this case, she does not need help. She may sometimes receive a good income which may provide for her needs for a few months. When she uses up this income and is back to poverty, she may be helped during her periods of need. In fact, she must be helped, if not from zakat, then from another charity or sadaqah [voluntary charity].
And God the Almighty knows best.