Some poor people are simply people with very low intelligence. Not low enough to be in some institution, but low enough to do stupid things that a) make them poor, and b) make them unlikely to use contraceptives however free and accessible, even in the case they don't really want to have children.

Possible solution: Provide free food to people. With contraceptives. Make this fact known. Expect outrage (not from the people who will eat the food, but from people who will use this opportunity to signal their moral superiority over you).

Some people have inherited a culture that does not value education and long-term planning. Historically, this culture came to my country centuries ago from a different place (where perhaps, in given time, it was not as disastrous as it is here and now). But looking at individuals, this culture comes from the previous generations.

Possible solution: If I had one, I would be going for my Nobel price now, instead of commenting on LessWrong. The solution would have to be not just realistic, but also politically acceptable, which is almost impossible given the connotations. (You know who else proposed solutions? Nazis did.)

If we remove the "politically acceptable" part, then a possible solution would be mandatory education for one generation. And by mandatory education I do not mean the usual mandatory school attendance, which we already have, and which is largely without an effect for given groups, because they easily undermine it. The pressure to learn from school would have to be greater than the pressure to not learn from home. Which would require some serious brainwashing and possibly isolation from home for longer periods of time. (Note: The "brainwashing" could be realized gently, e.g. through movies that depict smart and diligent heroes, and stupid contemptible villains.)