  We prove the existence and uniqueness of continuous solutions to the complex
Monge-Amp\`ere type equation with the right hand side in $L^p$, $p>1$, on
compact Hermitian manifolds. Next, we generalise results of Eyssidieux, Guedj
and Zeriahi \cite{EGZ09, EGZ11} to compact Hermitian manifolds which {\em a
priori} are not in the Fujiki class. These generalisations lead to a number of
applications: we obtain partial results on a conjecture of Tosatti and Weinkove
\cite{TW12a} and on a weak form of a conjecture of Demailly and Paun
\cite{DP04}.
