I'm glad we agree that a sixteen year old single mother needs assistance whether or not she is at home, at school, or working.
But why do we agree assistance is necessary in the first place? Because, we have both agreed, that there is a societal need to provide assistance to those in poverty so as to provide social stability (again, for the purposes of this argument).
If this girl is on assistance already, that means there is minimum level of support she is unable to provide for herself, even if she is working (at, what we assume, is a minimum wage job in undesirable circumstances that only a dropout can get). So, when you say things like this:
...you are advocating that we replace some of the need that the socially stabilizing assistance is already working to reduce, i.e. increase her need. Assistance should only be enough to provide social stability, right? I am assuming that means basic needs (please correct me if this part is wrong).
So, if we are already topping up her income to provide for basic needs, in either situation as you readily agreed, then what purpose does it serve to fine her and put her back under her level of need? If the point of welfare spending is social stability through lessening individual need, wouldn't an increase in individual need lead to a reduction in social stability? If stability is the goal of the program in the first place, why would we want to directly counteract that goal?