  Assuming a variant of the Hardy--Littlewood prime $k$-tuple conjecture, we
compute mixed moments of the number of primes in disjoint short intervals and
short progressions respectively. This involves estimating the mean of singular
series over any product of lattices, which is of independent interest. As a
consequence, we establish the convergence of both sequences of suitably
normalized primes to a standard Poisson point process.
