What would happen if Superman and Supergirl had a child?
Okay first of all that would never EVER happen because Superman & Supergirl are COUSINS so a romance between them would be about as bad as a romance between a brother & sister and of course Superman is married to Lois Lane and they now have a son. But suppose for a moment they were not cousins and they did have a child. Quite frankly it would be almost about the same as when Jor-El & his wife had Kal-El on Krypton but since Superman & Supergirl are on Earth their child would of course inherit their Kryptonian superpowers and no doubt become the next Superman or Supergirl depending on what the gender is. But like I said that would never happen since they are cousins and Superman already has Lois Lane and they now have a Superman Jr in the family.
There’s no way that Superman and Supergirl have not studied science, including the science of genetics. They probably have a better grasp of it than any actual human. In the Silver Age, Supergirl tried to set up Superman with her analogues from other universes and solar systems – so the genetics might not have to be all that different for it to be okay to produce a child. Consider this also: Superman and Supergirl as Kryptonians under a yellow sun on Earth are fundamentally always in perfect health. That may include genetic abnormalities. If Superman or Supergirl had genetic abnormalities prior to arrival on earth, they might not know about them. It’s entirely possible that any offspring word correct genetic abnormalities in utero. Even if this were not the case, The most likely scenario for a child would be using Kryptonian science to select a fetus without deleterious inbreeding or to remove the defective genes. Doctor Fate or Zatanna could use a magic to change the genes. Even the universe with a minimal Kryptonian presence contains General Zod as a gene donor – a Krisp-Ur, as it were. With a child conceived under the influence of red kryptonite not switch back to whatever they would have been when their parents switched back because they already were their default?
Given the universe that Superman and Supergirl live in, I don’t think it would be as bad as in the real world.