We claim that if $n = 23k + 20$ with $k$ a non-negative integer then the equation has no solution over the integers. Let's assume that there is one and get a contradiction. Indeed, by Fermat's little theorem,
$$x^2 + y^{11} \equiv n + z^{2022!} \equiv 20 + (0 \text{ or } 1) \equiv 20 \text{ or } 21 \pmod{23}$$
and hence
$$x^2 + (-1, 0 \text{ or } 1) \equiv 20 \text{ or } 21 \pmod{23}$$
because $-1, 0$ and $1$ are the only congruence classes modulo $23$ such that its squares are congruent to $0$ or $1$ modulo $23$.
It follows that
$$x^2 \equiv 19, 20, 21 \text{ or } 22 \pmod{23}$$
which is a contradiction because neither of them is a quadratic residue as can be seen by direct inspection.