If someone suspects they need non-emergency medical care but they have little money, what can they do?

Free clinic or CHC. If you live near a larger city or university town, they may have a free clinic. There are also federally-funded community health centers which see low-income and uninsured patients at a discounted rate. Some docs in private practice may be willing to see you at a discount - ask around. Try calling your local county health department to find out what resources exist in your community.
Sliding scale. Difficult situation. In general an urgent care center would be the choice for non-emergent conditions. Try calling the center and see if they do "sliding scale payments." Depending upon how much you have / money you make there would be a reduced fee. Good luck in your search.