  Let $f$ be a transcendental meromorphic function defined in the complex plane
$\mathbb{C}$. We consider the value distribution of the differential polynomial
$f^{q_{0}}(f^{(k)})^{q_{k}}$, where $q_{0}(\geq 2), q_{k}(\geq 1)$ are
$k(\geq1)$ non-negative integers. We obtain a quantitative estimation of the
characteristic function $T(r, f)$ in terms of
$\overline{N}\left(r,\frac{1}{f^{q_{_{0}}}(f^{(k)})^{q_{k}}-1}\right)$.\par Our
result generalizes the results obtained by Xu et al. (Math. Inequal. Appl., 14,
93-100, 2011) and Karmakar and Sahoo (Results Math., 73, 2018) for a particular
class of transcendental meromorphic functions.
