In "Thank you, M'am," the narrator never explains why Roger asks to go to the store for Mrs. Jones, so it is up to us to analyze clues in the story to try to understand his motivation.
As Mrs. Jones is cooking him dinner, Roger asks,
Do you need somebody to go to the store?
Mrs. Jones says no, unless he wants some "sweet milk" for himself.
There are several plausible reasons why Roger would ask to go to the store. He has just tried to rob Mrs. Jones, and this powerful, intimidating woman has dragged him to her apartment. So one explanation would be that he wants to get away from her. However, the story makes clear he already has ample opportunity to do that. The door to her apartment is open, and he realizes he could make a "dash" for it and "run, run, run, run, run!" Given that he doesn't need to go to the store to escape from her, we might look for another explanation.
The story says that, after considering making a run,
the boy took care to sit on the far side of the room where he thought she could easily see him out of the corner of her eye, if she wanted to. He did not trust the woman not to trust him. And he did not want to be mistrusted now.
Roger has decided he wants to earn Mrs. Jones's trust. He feels comforted and safe that she is treating him as a mother would a son. He has nobody at home to take care of him, but now this woman has made him wash his dirty face and is making him a dinner of ham and green beans. This would suggest that rather than trying to get away, the boy wants to prove to Mrs. Jones that he is trustworthy. If he runs an errand for her with her money and comes back with whatever she wanted him to buy, that will prove to her he is honest.
The point of the story is that Mrs. Jones's tough love is having a healing effect on the neglected child. By treating him as a person worthy of respect and care, even though he tried to rob her, she instills in him the idea that he should be better behaved. Hughes is saying that people who are cared for and trusted become caring and trustworthy themselves.