You can't budget your way out of being poor. Most actually poor people (as opposed to people who have a substance abuse problem) I know have a very good grasp of their budget as they are constantly shifting money around figuring out which bills they have to pay and which bills they can put off.
You get out of being poor by getting more money. Period. Nothing else works.
Yes, more money doesn't guarantee you get out of being poor, and we all know people who got a windfall and then were worse off than before.
However, insufficient money absolutely does guarantee that you will be poor indefinitely.