Suppose a polynomial time machine that has an access to a polynomially long string of bits independent on the input. On average, it's impossible to compress this string to a subpolynomially long string using a deterministic polynomial time compression algorithm.
As far as I understand, such a machine would be a $BPP$ machine, even though the source of bits is not necessarily a random number generator.
In this case the amount of independent bits is polynomial. I suppose, if we allow only a constant amount of independent bits (e.g. how it happens with most existing PRNGs using system time - a fixed length independent string - for a seed) then the PRNG gives no extra computation power. However, can it be above constant and still keep the machine equal to a polynomial time deterministic machine?