PRNG output with truly random noise
Usually what you call "information theoretic hardness" is denoted as statistical indistinguishability. And it is pretty much black and white only: Two distributions are distinguishable or not.

When you ask "how much harder", you already drift into the area of computational indistinguishability. There are plenty of ressources for that keyword, including most books about introduction into cryptography.

So the actual answer to your question would be "If the distribution of p is distinguishable (which is the case for any $p\neq1/2$), and the distribution of the PRNG is distinguishable, then your construction is distinguishable (from a truly random distribution)"