What is the WPA and why won't Mr.Cunningham work for it? What does this reveal about his character?
The Cunninghams are poor but they are too proud and self-respecting to take handouts. At Scout's first day of school, Walter has no lunch and no money. Miss Caroline offers him a quarter for lunch and he refuses. Scout gives Miss Caroline clarification:
"The Cunninghams never took anything they can’t pay back—no church baskets and no scrip stamps. They never took anything off of anybody, they get along on what they have. They don’t have much, but they get along on it."
Scout also mentions that Atticus represented Walter Cunningham Sr. regarding his entailment. Walter Sr. could not repay Atticus with money, but he did eventually pay him back with goods: wood, hickory nuts, holly, turnip greens, etc. (Part of the land Walter farmed was entailed, meaning that he was not the clear owner of it.)
The part of the land that Walter does own is mortgaged "to the hilt." Therefore, Walter spends most of his earnings paying interest on that mortgage. He pays it because he wants to keep the bit of land that he does fully own. He has a sense of pride about this.
Walter's pride is significant here. The WPA, Works Progress Administration, was part of FDR's "New Deal." The WPA was designed to help people find jobs. Walter might be too proud to take a job from the government. He may also decline the WPA job because he doesn't want to be politically affiliated with FDR's government policies. (He can therefore vote as he pleases, according to Scout). But he also simply wants to keep his own land. This is probably the main reason he rejects the WPA.