To cry poor is to whine, complain, or make excuses about how you're very poor or how your life is hard because you're very poor.

For example, you're crying poor if you say, "Uhh, I really really want extra cheese on my pizza, but I sooo can't afford it right now," or "I could look as trendy as SHE does if I could shop at those fancy stores too," or "I'm so stressed right now; I have to choose between upgrading my awful computer and paying my cell phone bill on time because there's no way I can do both."

Crying poor is something that Jesse's mother does because she really is poor and her life really is hard, but crying poor is still considered whiny, unattractive behavior. 

The narrator tells us in Chapter 8 about how she does this:

His mother always cried poor, but she put a lot of thought and as much money as she could scrape together into making sure she wouldn't be embarrassed by how her family looked.

What this means is, normally, if Jesse's sisters asked their mother for new clothes, she'd usually say something like, "No way. You know we can't afford that, with times being so hard." She could just be honest and straightforward with her girls, but by adding the touch of complaint to her words and trying to make her daughters feel guilty or ashamed for asking, the mother is crying poor.