We explore the cohomological structure for the (possibly singular) moduli of \(\mathrm{SL}_n\)-Higgs bundles for arbitrary degree on a genus g curve with respect to an effective divisor of degree >2g-2. We prove a support theorem for the \(\mathrm{SL}_n\)-Hitchin fibration extending de Cataldo's support theorem in the nonsingular case, and a version of the Hausel-Thaddeus topological mirror symmetry conjecture for intersection cohomology. This implies a generalization of the Harder-Narasimhan theorem concerning semistable vector bundles for any degree. Our main tool is an Ng\^{o}-type support inequality established recently which works for possibly singular ambient spaces and intersection cohomology complexes.