The conjecture PSQ 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, and http://oeis.org/A005377 for a list of prmes not of the form $p+x^2$ with $p$ prime and $x$ a *positive* integer.

Concerning your second question on *uniform explanations*, you may consult  Conjecture 2.1 of my paper [Conjectures on representations involving primes][1] published in 2017 for a **General Hypothesis** on representations involving primes.

PS: I don't think it is easy to pose new nice conjecures on primes.

[1]: https://doi.org/10.1007/978-3-319-68032-3_20