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 strucutre.
In fact, Fujiki did work 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 $p=0$, which are the so called du Bois singularities.