What you are talking about doesn't sound like a behavioral economics question. Behavioral economics would concern itself more with why a person wouldn't exploit a poorly constructed welfare system. What you are asking is perhaps more related to market failure and how non-excludability and non-rivalry lead to a rational person taking advantage of such a welfare system. I say the system has to be poorly designed, or else the desired effect of helping children would naturally be benefitted if having more children resulted in higher welfare, which would in term help parents to care for those additional children. The example you used was a result of people claiming dependents they did not have ("other people's children") and welfare was given to people who had no claim to it.
This is resolved by amending one or more aspects of the market failure of the public good, such as rivalry: having multiple agencies of the government provide this welfare and cutting the one that does a worse job, leading to a greater incentive for agencies to gather accurate data; or non-excludability: requiring more stringent measures for receiving said welfare, such as children's birth certificates, school report cards etc.
It is interesting as we see a possible route for amending this market failure, perhaps subsidizing school costs instead of a lump sum welfare for the parent, is already being addressed by our current school system, albeit in a poor way (providing education for a sizable portion of students future income in terms of school loans and resulting interest payments).
A weak mechanism will always be exploited in the view of economists as the exploiters are homo economicus, a perfectly rational and utility-maximizing being.