The local health department offers many classes for the community free of charge. They have nutrition programs, evaluation staff that can do things like screening for sensory issues and even ADHD. They're not medical staff but they can do some simple screening and give you some good ideas to try. They do pregnancy testing, give flu shots and kids vaccines, they do stop smoking clinics, they do VD testing, birth control, and they'll even help families who need new car seats when they have funds.
Some of the programs are income based but they are not for extremely poor people only. The income qualifications are higher than most of the state assistance programs.
WIC is a program that gives food to pregnant women and formula to infants. That program is based on income but it's not for really really poor people only. The income qualifications for WIC are not the same as food stamps or welfare. They are high enough that working families can have WIC too.
Overall the county health department is an underused resource for many people and especially women.
Free clinics would be something else entirely. This is not a doctors office you go to for medical care. This is a center for people to go to for programs that would help them in rather short term endeavors.
All you need to do is find out what county you live in then google for that county health department. Then scroll through and see what programs they offer.
I've gone to many classes there. I took a nutrition class that was a couple of hours per week and lasted a month or so. We got a few things from local farmers markets so we could try new foods and got recipes that day to use those particular items.
The health department is not a medical office nor is it a low income only place.
Here is a link to all the health departments in OK. The statement they put on here is this.
"Each department offers a variety of services, such as immunizations, family planning, maternity education, well-baby clinics, adolescent health clinics, hearing & speech services, child developmental services, environmental health, and the SoonerStart program."
http://www.ok.gov/triton/modules/health/map/county_map.php