Fix an odd prime $p$. Let $\alpha = (\alpha_0,\dots,\alpha_k)$ be a solution to the congruence $\sum_{i=0}^{k} \alpha_i^2 \equiv x \mod p$. Now consider the number $N_\alpha$ of solutions to the simultaneous congruences $$ \sum_{i=0}^k \beta_i^2 \equiv y \mod p \quad \text{ and } \quad \sum_{i=0}^k (\alpha_i + \beta_i)^2 \equiv z \mod p. $$

Based on some small computations, I am finding that $N_\alpha$ does not depend on $\alpha$, i.e., it seems there exists a constant $c$ (depending on $x$, $y$, and $z$) such that $N_\alpha = c$ for any solution $\alpha$ to the first congruence.

Does anyone know whether this is true in general? Or can someone provide a useful reference?