  We consider small solutions of quadratic congruences of the form
$x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$, where $q=p^m$ is an odd
prime power. Here, $\alpha_2$ is arbitrary but fixed and $\alpha_3$ is
variable, and we assume that $(\alpha_2\alpha_3,p)=1$. We show that for all
$\alpha_3$ modulo $p^m$ which are coprime to $p$ except for a small number of
$\alpha_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$
to the congruence $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$ with
$\max\{|x_1|,|x_2|,|x_3|\}\le N$ holds if $N\ge q^{11/24+\varepsilon}$ and $q$
is large enough. It is of significance that we break the barrier 1/2 in the
above exponent. If $q$ is restricted to powers of a fixed prime $p$, we obtain
a slight improvement of this result using the theory of $p$-adic exponent
pairs, as developed by Mili\'cevi\'c, replacing the exponent $11/24$ above by
$11/25$. Under the Lindel\"of hypothesis for Dirichlet $L$-functions, we are
able to replace the exponent $11/24$ above by $1/3$.
