In the Metropolis-Hastings algorithm, depending on the current state $x$, I have a distribution $\rho_x$ and I want to use a sample from $\rho_x$ as the proposal in the next iteration. (I guess technically this means to start the Metropolis-Hastings algorithm afresh with a new proposal kernel and for only one single iteration.)
Now the crucial thing is that I cannot sample from $\rho_x$ directly, but also need to run the Metropolis-Hastings algorithm to obtain a sample $y$ from it.
What I worry about is that $y$ is not exactly distributed according to $\rho_x$; since the Metropolis-Hastings algorithm only guarantees that the limiting distribution is $\rho_x$. So, am I allowed to use the density of $\rho_x$ inside the "outer" Metropolis-Hastings algorithm as the density for the proposal kernel or does this lead to any issues? If not, is there a reference on that topic which shows that we are allowed to do that?