Can we find an explicit solution of this Poisson equation?

Question

Let $\kappa$ be a Markov kernel with invariant measure $\pi$ and $$A:=\kappa-\operatorname{id}$$ denote the corresponding (discrete-time) generator of $\kappa$. Let $c>0$ and $$r:=\frac cp$$ where $p$ is the density of $\pi$ with respect to some reference measure. Given a $\phi\in\mathcal L^2(\pi)$, I want to find $\eta\in\mathcal L^2(\pi)$ such that $$-(A-r)\eta=\phi\tag2.$$ I know that in general the solution of such a Poisson equation is given by the potential operator. However, if $\kappa$ is actually equal to the measure $\pi$ itself, is there an easier solution available?