  We consider the complex Monge-Amp\'{e}re equation on complete K\"{a}hler
manifolds with cusp singularity along a divisor when the right hand side $F$
has rather weak regularity. We proved that when the right hand side $F$ is in
some \emph{weighted} $W^{1,p_0}$ space for $p_0 > 2n$, the Monge-Amp\'{e}re
equation has a classical $W^{3,p_0}$ solution.
