On distribution of elements of subgroups in arithmetic progressions modulo a prime

Let $\mathbb F_p$ be the field of residue classes modulo a large prime number $p$. We prove that if $\mathcal G$ is a subgroup of the multiplicative group $\mathbb F_p^*$ and if $\mathcal I\subset \mathbb F_p$ is an arithmetic progression, then $|\mathcal G\cap \mathcal I| = (1+o(1))|\mathcal G|\kern 1pt|\mathcal I|/p + R$, where $|R|<\bigl (|\mathcal I|^{1/2}+|\mathcal G|^{1/2}+|\mathcal I|^{1/2}|\mathcal G|^{3/8}p^{-1/8}\bigr )p^{o(1)}$. We use this bound to show that the number of solutions to the congruence $x^n\equiv \lambda \pmod p$, $x\in \mathbb N$, $L<x<L+p/n$, is at most $p^{1/3-1/390+o(1)}$ uniformly over positive integers $n$, $\lambda $ and $L$. The proofs are based on results and arguments of Cilleruelo and the author (2014), Murphy, Rudnev, Shkredov and Shteinikov (2017) and Bourgain, Konyagin, Shparlinski and the author (2013).