What are the best breathable cover masks available? ?

Double check the mask policy where you are. 

I am a "mask required" area. But, I am not required to wear a mask when outdoors. When not wearing a mask outdoors, I am required to keep distance from others.

The same rule might be in effect where you are.

If you are required to wear a mask outdoors, see if you are allowed to wear one with intake valves.  Also, check to see if face shields can be allowed instead of masks. But, even if allowed, you might find running in a face shield also doesn't work for you.

(DO NOT wear a mask with exhaust valves! A mask with an exhaust valve can be worse for other people than wearing no mask.)

If you don't have to wear one outdoors, I'd recommend not wearing one.  Something thin would be key, if you must use one but it's bad enough trying to breathe when it's really hot out and you're tired. I can't imagine adding a mask to my running attire.