Are all integers up to $x$ but possibly $O_{\varepsilon}(x^{\varepsilon})$ the sum of $a$ squares and $b$ primes with $a+b\leq 3$?

Question

This question is related to https://math.stackexchange.com/questions/3710032/conjecture-all-but-21-non-square-integers-are-the-sum-of-a-square-and-a-prime.

We know since Lagrange that every natural integer is the sum of $4$ squares. Euler proved Fermat's assertion stating that primes of the form $4n+1$ are the sum of two squares in a unique way up to order and signs. On the other hand, Goldbach weak conjecture, stating that every odd integer is the sum of three primes, was proved by Helfgott in 2013. Of course, a proof of the strong Goldbach conjecture would provide an affirmative answer to the following question:

can one, with current technology, prove that every integer up to $x$ but $O_{\varepsilon}(x^{\varepsilon})$ is the sum of $a$ squares and $b$ primes with $a+b\leq 3$? Note that Helfgott's result prevents the inequality from being strict.

Answer 1

Linnik (1960) proved that every sufficiently large positive integer is the sum of a prime and two squares, confirming a conjecture of Hardy-Littlewood (1923). Moreover, he gave an asymptotic formula for the number of representations. See

Y. V. Linnik, An asymptotic formula in an additive problem of Hardy-Littlewood (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960), 629–706.