I believe that people should not have children until they have attained a minimally adequate level of education, are working and have a life partner. I do not believe that children should be conceived when the likely consequence is that parents and child will be welfare dependent. But of course, the world is not perfect. Unplanned pregnancies do happen and the circumstances of parents can change.
If a single woman has a child, the first person who should be asked to pay is the father. But there are also cases where relying on the father to pay is not possible. Maybe the father is unknown or a tourist. Maybe the father is himself on welfare.
My view is that in such a situation, the government should pay welfare to the mother. But, and this is the difference to the current system, only for one child (except in special cases, for example twins). The father (if possible) and the mother would each have a family planning consultation with a qualified professional. It should be made clear to the mother that there will be no extra money paid for a second child if welfare dependency was predictable at the time of conception. If the father is dependent on welfare (which he probably would be if he is not in a position to pay maintenance to the mother), he would have his benefits drastically reduced if he were to get a second woman (or the same woman again) pregnant in a situation where it is predictable that he can't pay.
I think that strikes a reasonable balance between the harshness of the pre-1973 era where girls and women who slipped up just once were expected to give up their child for adoption and the current system where some welfare-dependent people have child after child and the assumption that the government must pay is rarely challenged.
There all sorts of situations where someone could end up a single parent of multiple children due to unforeseen circumstances. It is entirely proper that the welfare system assists such parents.