If your kids have to live on the street if you take that part-time job, it's not only short-sighted to stay on welfare, it's the rational thing to do even in the long term.

Even more rational is finding an unofficial source of income, which is what people in this situation often do. At scale, this may create a wrong impression that levels of welfare are adequate to guarantee the basics.

Yes, and worse, the incidents of unreported income like that also create a narrative to be spoon-fed to the public, that “welfare cheats” are taking advantage of everybody’s goodwill. I’m sure it actually happens. Everything gets abused by some. But people who are doing what you describe are usually not doing it out of greed, but to work around that broken system.