Can a PRNG be made into a CSPRNG by hashing the outputs?
No, the question's construction is not guaranteed to turn a PRNG into a CSPRNG. In some cases, the hashing worsens the generator. In more, undesirable characteristics remain detectable after hashing. There are better methods to build a CSPRNG from a hash.



The question's construction, hereafter WBCSPRNG for Would-Be Cryptographicaly Secure Pseudo RNG, is a PRNG with the same input characteristics and state size (within a few bits depending on how that's counted) as the original PRNG. It has a period demonstrably not smaller than the PRNG by a large factor (at worse 1024/256=4) under the hypothesis that the hash behaves as a random oracle. I do not see anything else comforting that can be said unless we add some hypothesis about the PRNG.

In particular, we can devise a PRNG, with a large state and long period, such that the WBCSPRNG fails most existing PRNG test suites, when the original PRNG consistently passes them. Start from a good CSPRNG, and modify it so that after having output 1023 bits, the next bit is produced by hashing these and a zero bit, summing the 256 bits yielding $b\in[0,256]$, outputting zero if $b&lt;128$, one if $b&gt;128$, and the low-order bit of the hash otherwise.

Our constructed PRNG is no longer a CSPRNG, but still pass any pre-existing RNG test (no pre-existing RNG test will catch how the extra bit is generated). Yet the WBCSPRNG is biased towards zero, and most RNG test suites will detect that.



The above counterexample is maliciously constructed, and in a way that  needs to know the hash, which we could forbid. That's not good enough: a whole class of accidental defects are amplified. For example, if a PRNG has the undesirable characteristic that the first 1024-bit block of each 242-bit output block is identical, the WBCSPRNG has the characteristic that the first 256-bit block of each 240-bit output block is identical, which is likely to cause issue sooner.

The hypothesis when academically evaluating the security of a CSPRNG is that its design is public, here including the WCSPRNG's hash step and the PRNG. If the PRNG has the characteristic that a class of 1024-bit blocks are likely to appear in its output, that again can be detectable in the WCSPRNG output, after testing 4 times less bits.



The method has no practical interest as a way to construct a CSPRNG, for we do not need a PRNG to make a CSPRNG out of a hash such as SHA-512/256, and one more efficient than the question's one: we hash the seed input, set a counter to that, then to produce 256 bits of output we hash the state then increment it.