Let \(f(x)\) be a polynomial of degree \(n\) and \(b\) be a nonzero integer. Also, if \(P\) is a polynomial, let \(M(P)\) denote its Mahler height. The authors consider the equation \[ f(x)= by^z, \quad x,y,z \;\text{integers and} | y| >1, \quad z>1. \] They prove that if \(f\) has at least two distinct zeros, then \[ z < cM^{3n}\log^3| 2b|, \] where \(c\) is an effectively computable constant depending only on \(n\). The proof depends on two lemmas: One on linear forms in logarithms by P. Philippon and M. Waldschmidt [New advances in transcendence theory, Proc. Symp., Durham/UK 1986, 280-312 (1998; Zbl 0659.10037)], and the other on an estimate of the logarithmic height for a system of fundamental units due to Y. Bugeaud and K. Györy [Acta Arith. 74, 67-80 (1996; Zbl 0861.11023)].