I am considering a situation where an adversary does not have access to the $n$-bit output string of a PRNG, but instead receives a noisy version of it, where each bit of the string is flipped with probability $p$. We assume that this noise which causes the bits to flip is truly random.
What can we say concretely about the difficulty of distinguishing this noisy PRNG output from a truly random string? Of course, for $p=0$ we recover the standard scenario for the PRNG and the problem is as hard as solving the underlying computational problem, and for $p=\frac12$ the string is truly random. But what about intermediate values of $p$? Does the distinguishing problem become significantly more difficult even if $p\ll 1$?