Damp patches in basement floor
Moisture is coming up through the concrete from below. This is normal in the wet season depending on your area.

NEVER put carpet on a basement floor. It is just a really bad idea. It is a sponge for everything from a leaking water heater, to moisture from humidity in the summer to pet urine, etc. It also hides problems like this until they become moldy and unhealthy.

Clean and patch the floor as best you can and stain it (best) or paint it with epoxy basement paint (better than carpet). If you get moisture in the future it will not damage stain but it may make a small area of paint peel.

Any way you go the moisture is going to continue coming up through the floor unless you hammer up all the concrete and re-pour it with a plastic vapor barrier under it. Modern building codes require a vapor barrier under basement floors to prevent these problems.

Good luck with your project!