The answer is that if $X$ is compact complex space of class $C$ in Fujiki's sense (i.e. if it is dominated by compact Kaehler manifold) then $H^*(X)$ carries a natural mixed Hodge structure. This is gotten by using resolution of singularities to express the cohomology of $X$ in terms of the cohomology of simplicial Kaehler manifold, and then applying the methods of Deligne's Hodge II & III (as Shengao points out) to give this a mixed Hodge structure.
In fact, Fujiki worked this out long ago in "Duality for Mixed Hodge structures..." RIMS 1980.
I realize there was more to your question. The precise relationship between $H^*(X,\mathbb{C})$ and $H^q(X,\Omega_X^p)$ for singular spaces is, to put it mildly, complicated. At least in the algebraic category, Du Bois has shown that there exists objects $\tilde \Omega_X^p$ in the derived category, such that $$H^i(X,\mathbb{C}) = \bigoplus_{p+q=i} H^q(X,\tilde \Omega_X^p)$$ There are maps $\Omega_X^p\to \tilde \Omega_X^p$. The question of when these are isomorphisms is not well understood; except perhaps for $p=0$, where isomorphism characterizes the so called Du Bois singularities.
I thought I'd add a few rather speculative comments. Our answers (mine and Sándor's) are taking this in somewhat homological direction -- a lot of Hodge theory tends to get that way. This is perhaps a bit unfortunate, because for many the initial attraction to the area stems from its analytical aspects. I've often wondered *is there a purely analytic approach to mixed Hodge theory?* I remember having a conversation with Saper, long ago, who thought it might be feasible to do this as some sort of weighted $L^2$ cohomology for a suitably chosen metric on the smooth part. I'm being purposely vague here. Anyway, I don't think anyone has ever carried out anything like this. It would be really interesting if someone did.
Let me try to make this a bit more precise, although it will still be pretty vague. Suppose that $X$ is a singular compact complex space which can be blown up along the singular locus $X_{sing}$ to a Kaehler manifold. Let $U = X-X{sing}$. Then there are various choices of complete Kaehler metric on $U$ for which the space of harmonic forms satisfying $\int \alpha\wedge *\alpha < \infty$ coincides with intersection cohomology $IH^i(X)$. This gives a pure Hodge structure on it by the Kaehler identities. Although this is not quite what I'm asking for, it points in the right direction (note that I'm pretty sure that $IH^i(X)$ is a pure subquotient of $H^i(X)$). The question is how to modify this picture so that one gets the mixed Hodge structure on $H^i(X)$? A weaker question is how to describe the pure subquotients $W_kH^i(X)/W_{k-1}H^i(X)$ by analytic means? One can ask something like this for the Du Bois complex as well. I don't want to get much more specific. But perhaps I can point out, at least one useful reference: Saper, "$L^2$ cohomology on Kaehler varieties with isolated singularities" JDG (1992).