Does foundation primer cover up redness?

It depends on the primer. Their are "tinted" ones that can reduce redness, undereye circles etc. I use Tarte's Clean Slate, it doesn't break me out but it's kind of pricey. If your wanting something a little cheaper I really like, Revlon Beyond Natural Smoothing primer and Sally Hansen Natural Beauty Luminizing primer. Those are both clear though, NYX makes 2 different primers(main differences are container and one is HD while the other is not) and they both come in Green. The Green will counterbalance the Redness and you won't be red nor green :)