Heritage's political agenda aside, the relationship between 'amount of time spent on welfare' and 'child poverty' is not necessarily spurious - IF, the former is a proxy for the duration of living in poverty. People with long history of poverty are less likely to teach their young what it takes to be admitted to the middle class: the importance of formal education, and the accumulation of 'social capital' (i.e. learning the right mode of speach, dress code, and manners, getting to know the right people, etc.
Lest the pc crowd on this list starts screaming "culture of poverty" - this argument does not have to lead to the "family values" and "personal responsibility" mantra. It can be used to argue for an effective public education system that can overcome the shortcomings of nuclear families (whatever their roots) by teaching the disadvantaged kids the skills that their nuclear parents cannot.
It would be a very clever thing to use the Heritage's own findings to that end, no? As an ancient Sufi proverb has it, "a fool tries to convince me with his arguments, a sage - with my own."