The conjecture is essentially not new. In 1923 Hardy and Littlewood [Acta Math. 44(1923), 1-70] conjectured that every large integer, not being a square, may be expressed as the sum of a prime and a square. See also http://oeis.org/A020495 for the list of non-square positive integers which are not of the form $p+x^2$ with $p$ prime.
I don't think it is easy to pose new nice conjecures on primes.