Diophantine equations in the primes

Let p = (p(1), ... , p(r)) be a system of r polynomials with integer coefficients of degree d in n variables x = (x(1), ... , x(n)). For a given r-tuple of integers, say s, a general local to global type statement is shown via classical Hardy-Littlewood type methods which provides sufficient conditions for the solubility of p(x) = s under the condition that each of the x(i)'s is prime.