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).

as opposed to complex manifold)", so a complex space may possibly be singular and noreduced. [the standard defn of a complex space is: a loc ringed space loc isom to the quotient of the sheaf of holomorph funct's on a domain of $\mathbb{C}^n$ by a sheaf of ideals] – Qfwfq Mar 13 '11 at 15:53