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We introduce the $L^p$ Poisson-Neumann problem for an uniformly elliptic operator $L=-\rm{div }A\nabla$ in divergence form in a bounded 1-sided Chord Arc Domain $\Omega$, which considers solutions to $Lu=h-\rm{div}\vec{F}$ in $\Omega$ with zero Neumann data on the boundary for $h$ and $\vec F$ in some tent spaces. We give different characterizations of solvability of the $L^p$ Poisson-Neumann problem and its weaker variants, and in particular, we show that solvability of the weak $L^p$ Poisson-Neumann probelm is equivalent to a weak reverse H\"older inequality. We show that the Poisson-Neumman problem is closely related to the $L^p$ Neumann problem, whose solvability is a long-standing open problem. We are able to improve the extrapolation of the $L^p$ Neumann problem from Kenig and Pipher by obtaining an extrapolation result on the Poisson-Neumann problem., Comment: 49 pages