There are people that have skin breakouts whilst wearing a mask. In Melbourne we have to wear masks unless there is an exceptional one not to be wearing one. A skin breakout is not one of those.
People have been fined for not wearing a mask.
If you have a skin problem (ie a rash or acne) whilst wearing a mask, I would say these things to help alleviate the problem:
Have a good diet full of vegetables, grains etc. Make sure that you are getting enough fiber
Drink lots of water. Lots of water helps to hydrate your skin.
Wear a fabric mask if you can. Fabric is a lot easier to breathe in. Those paper ones can easily dry your skin out. You can also wear a scarf or a buff as a face mask. But some fabric masks might be made out of harsh materials and some people cannot wear wool. So find what works for you. Most people can wear a cotton mask.
What soap or laundry detergent you wash your mask in matters. If you use a soap or detergent that is harsh on your skin, then you’ll have some skin problems. I love OMO laundry detergent as its not harsh on sensitive skin. I also have this soap at home that is good for sensitive skin.
Take care of your skin. Make sure that you are wearing plenty of moisturiser