What I've read says that almost everywhere in the poor areas of the third world, basic birth control is easily available and free, and even the poorest most uneducated know how it works. They just won't use it, for the religious/cultural reasons you noted. If you give them food for free without conditions it just enables them to do what they're already doing, making the problems worse.

I'm drinking coffee right now. If someone gives me a dollar without conditions it does enable me to drink coffee, and anything else I choose to do.