A low income person with child dependents and at least some income earned from working will be eligible for the child tax credit and the earned income credit. The amounts are on a curve that goes up and down so earning more money will help you get a larger credit up to a point, then earning more income after that reduces the credit. Exactly where the maximum credit will be depends on your income and other tax situations.
If the children and non-working parents are living with other relatives (grandparents, aunts and uncles) it can sometimes be more advantageous for the other relatives to claim the children as dependents. We would need to know a lot more details first though.
