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 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.