Abstract In this paper we obtain new upper bound estimates for the number of solutions of the congruence $$\begin{equation} x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in \mathcal{U}, \end{equation}$$ for certain ranges of H and | ${\mathcal U}$ |, where ${\mathcal U}$ is a subset of the field of residue classes modulo p having small multiplicative doubling. We then use these estimates to show that the number of solutions of the congruence $$\begin{equation} x^n\equiv \lambda\pmod p; \quad x\in \mathbb{N}, \quad L<x<L+p/n, \end{equation}$$ is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers n , λ and L , for some absolute constant c > 0. This implies, in particular, that if f ( x ) ∈ $\mathbb{Z}$ [ x ] is a fixed polynomial without multiple roots in $\mathbb{C}$ , then the congruence x f ( x ) ≡ 1 (mod p ), x ∈ $\mathbb{N}$ , x ⩽ p , has at most $p^{\frac{1}{3}-c}$ solutions as p → ∞, improving some recent results of Kurlberg, Luca and Shparlinski and of Balog, Broughan and Shparlinski. We use our results to show that almost all the residue classes modulo p can be represented in the form xg y (mod p ) with positive integers x < p 5/8+ϵ and y < p 3/8 . Here g denotes a primitive root modulo p . We also prove that almost all the residue classes modulo p can be represented in the form xyzg t (mod p ) with positive integers x, y, z, t < p 1/4+ϵ .