In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing forth and back transition probabilities for each Metropolis move.

Can I perform a Monte Carlo estimation with samples that are generated by "inflating" a sequence of independent samples?

As a simple 1D example, if there is a sequence $\{x_i\}$ of values $\{3, 8, 4\}$, then I would like to inflate it into a sequence $\{y^i_j\}$ that looks, e.g., like $\{2.9, 3, 3.1, \quad 7.8, 8.05, 8.07, \quad 4.2, 4.04, 4.01\}$. That is, I take an independent sample and inflate it into multiple samples by perturbing around it.

Given that the sampling probabilities are known for both independent sampling $p(\cdot)$ and for the perturbation of the sampled values $q(\cdot |x_i)$, how can one construct a proper Monte Carlo estimator based on such an "inflated" sequence $\{y^i_j\}$?

I have checked that the ad-hoc estimator

$I=\sum\frac{f(y^i_j)}{p(x_i)q(y^i_j|x_i)}$

does not lead to the correct estimation.