In a recent paper, we proved that for any large enough odd modulus \(q\in \mathbb {N}\) and fixed \(\alpha _2\in \mathbb {N}\) coprime to q, the congruence \( x_1^2+\alpha _2x_2^2+\alpha _3x_3^2\equiv 0 \bmod {q} \) has a solution of \((x_1,x_2,x_3)\in \mathbb {Z}^3\) with \(x_3\) coprime to q of height \(\max \{|x_1|,|x_2|,|x_3|\}\le q^{11/24+\varepsilon }\) for, in a sense, almost all \(\alpha _3\) , where \(\alpha _3\) runs over the reduced residue classes modulo q. Here it was of significance that \(11/24<1/2\) , so we broke a natural barrier. In this paper, we average the moduli q in addition, establishing the existence of a solution of height \(\le Q^{3/8+\varepsilon }\alpha _2^{\varepsilon }\) for almost all pairs \((q,\alpha _3)\) , with Q large enough, \(Q<q\le 2Q\) , q coprime to \(2\alpha _2\) and \(\alpha _3\) running over the reduced residue classes modulo q.
