The pattern is "I cannot afford NOUN". Here, the clause "for someone to do something" is a noun clause. The fact that it can be a noun clause has nothing to do with the word "afford" and your confusion is actually unrelated to that word. The thing confusing you is the fact that it actually is a noun clause. To convince you that it is a noun clause, I will use it as a noun clause in some other sentences: "For someone to swim here is dangerous" "For Jack to say that is outrageous" "I would like for the children to sing that song"

It is quite a common structure. For example, in a family with small children, maybe the mother wants to give up work and take care of the children. But they need two salaries to pay all their bills. So they can't afford for her to give up her job. In your example, we don't have any context. But it might mean something similar. For example, ideally, the children in the family would be able to live and act like children - just play or study, and have no responsibilities. But maybe this is a poor family and the children have to get part-time jobs to help. So they take on adult responsibilities and stop being children.