I think your question has two problems: first, it mentions
need. It presupposes that we can accurately know the inner motivations of every parent in each country. Second, you're asking for the efficacy of the support systems specifically (since you discarded the influence of prohibition laws with your
child labor is illegal comment). But no policy exists in isolation, so you cannot measure the effect of those systems in isolation.
In general, I can think of three different policies that all work to reduce child labor:
- a general ban on child labor, which obviously is a stick mechanism to prevent child labor
- compulsory education, which is another stick mechanism to make sure children do not work all day
- child benefit payments, which is a carrot mechanism to make sure parents have the means to feed and clothe their children
Then there's indirect policies like unemployment benefits, health insurance, and child abuse laws that also help parents to stay afloat and keep children out of labor. But the only measures we have are child labor incidence rate and school attendance rate. Neither of those measurements specifically measure the need for children to supplement the family income, nor do they measure the effects of the carrot policies alone.
So that's why I said your question is unanswerable: you're asking for the effects of specific policies on the decisions of parents, while the statistics will only give you the effects of the complete policy on the outcome of those parental decisions.
(edit: see https://ourworldindata.org/grapher/incidence-of-child-labour... for example. If you can find a way to compare the child labor policies of e.g. Argentina, Mexico, Serbia, Peru, and Nigeria and correlate them with this graph, maybe you'll be able to get closer to an answer to your question)