This is a follow up to my previous question (unfortunately closed as a duplicate). There the problem was to turn the Moore plane into a normal space by adding a single point. Brian M. Scott gave an answer to this specific problem years ago.
This got me thinking about a more general question: Can you add one point to any completely regular not normal space $X$ to obtain a normal space $Y$ of which $X$ is a subspace?
Note that for me a completely regular spaces and normal spaces are Hausdorff. The restriction to completely regular spaces is natural since every subspace of a completely regular space is itself completely regular, and normal spaces are completely regular. Since $Y$ is supposed to be normal, the subspace $X$ must be completely regular.
I know that if $X$ is locally compact you can take $Y$ to be the one-point compactification. However if $X$ is not locally compact the one-point compactification fails to be Hausdorff.