We consider a Markov kernel on a measurable space, satisfying a minorization condition and a modulated drift condition. Then we show that there exists a solution to the so-called Poisson equation whose norm can be bounded from above using the modulated drift condition. This new bound is very simple and can be easily computed. This result is obtained using the submarkov residual kernel given by the minorization condition. Such a bound allows us to provide new control on the weighted total variation norms of the deviation between the invariant probability measure $\pi_{\theta_0}$ of a Markov kernel $P_{\theta_0}$ and the invariant probability measure $\pi_{\theta}$ of some perturbation $P_{\theta}$ of $P_{\theta_0}$. From the standard connexion between Poisson's equation and the central limit theorem, a simple and computable bound on the asymptotic variance is also derived.