  Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$.
Let $\beta$ be a smooth real closed $(1,1)$ form such that there exists a
function $\rho \in \mbox{PSH}(X,\beta)\cap L^{\infty}(X)$. We study the range
of the complex non-pluripolar Monge-Amp\`ere operator
$\langle(\beta+dd^c\cdot)^n\rangle$ on weighted Monge-Amp\`ere energy classes
on $X$. In particular, when $\rho$ is assumed to be continuous, we give a
complete characterization of the range of the complex Monge-Amp\`ere operator
on the class $\mathcal E(X,\beta)$, which is the class of all $\varphi \in
\mbox{PSH}(X,\beta)$ with full Monge-Amp\`ere mass, i.e. $\int_X\langle
(\beta+dd^c\varphi)^n\rangle=\int_X\beta^n$.
