For any fixed \(k\in \mathbb{N}\), let \(f(x)\) denote the polynomial \(x^ k+ Qd\) with \(Q,d\in \mathbb{N}\). Put, for fixed \(Q\) and prime \(p\), \[ \rho(d,p)= |\{m \bmod p;\;m^ k+ Qd\equiv 0\bmod p\}|. \] We factorize \(f\) over \(\mathbb{Q}\) as \(f= \prod_{i=1}^{r(d)} f_ i\), where each \(f_ i\) is irreducible and has integral coefficients, then the prime ideal theorem implies that \(\sum_{p\leq x} \rho(d, p)\sim r(d)\cdot \text{li } x\), \(x\to \infty\). In the present paper the authors derive the upper bound \[ \sum_{Y\leq d\leq Y+H} \;\biggl| \sum_{X\leq p\leq 2X} (\rho(d, p)- r(d)) \biggr| \ll_{A,k, \varepsilon} {{X\cdot H} \over {\log^ A X}} \] uniformly for \(Q\leq Y^ A\), \(A>0\), provided that \(Y^{(1/2)+ \varepsilon} \leq H\leq Y\), \(\varepsilon>0\). This result has an interesting application concerning a theorem of Friedlander-Granville [J. Friedlander and A. Granville, Proc. R. Soc. Lond., Ser. A 435, 197-204 (1991; Zbl 0736.11049)] on irregularities in the distribution of primes represented by polynomials.