Let $F \in \mathbb{Z}[x, y]$ be an irreducible binary form of degree $d \geq 7$ and content one. Let $\alpha$ be a complex root of $F(x,1)$ and assume that the field extension $\mathbb{Q}(\alpha)/\mathbb{Q}$ is Galois.

We prove that, for every sufficiently large prime power $p^k$, the number of solutions to the Diophantine equation of Thue type $|F (x, y)|= hp^k$ in integers $(x, y, h)$ such that $\gcd(x, y) = 1$ and $1\leq h\leq(p^k)^\lambda$ does not exceed 24. Here $\lambda= \lambda(d)$ is a certain positive, monotonously increasing function that approaches one as d tends to infinity.

We also prove that, for every sufficiently large prime number $p$, the number of solutions to the Diophantine equation of Thue–Mahler type $|F (x, y)|= hp^z$ in integers $(x, y, z, h)$ such that $\gcd(x, y) = 1$, $z\geq 1$ and $1\leq h\leq (p^z)^{1/2-\kappa(d)}$ does not exceed 3984. Here $\kappa= \kappa(d)$ is a certain positive, monotonously decreasing function that approaches zero as $d$ tends to infinity.

Our proofs follow from the combination of two principles of Diophantine approximation, namely the Gap Principle, and the Thue-Siegel Principle.