Let $f(X)$ be an irreducible polynomial of degree $m\ge3$ with integer coefficients and unit leading coefficient, and let $\rho(n)$ be the number of solutions of the congruence $$ f(x)\equiv 0\pmod n; \quad 0\le X<n. $$ For certain classes of polynomials (in particular, for Abelian polynomials), the Dirichlet series $$ \sum_{n-1}^{\infty}\frac{p(n)}{n^s} \quad (\operatorname{Re}s>1) $$ has an analytic continuation to the left of the line $\operatorname{Re}s=1$. This allows us to obtain anasymptotic formula for $\sum_{n\le1}\rho(n)$ as $x\to\infty$, where the error term is better than that obtained on the basis of the modern theory of multiplicative functions.