Variable thickness sheet and homogenization-based topology optimization often result in spread-out, non-well-defined solutions that are difficult to interpret or de-homogenize to sensible final designs. By extensive numerical investigations, we demonstrate that such solutions are due to non-uniqueness of solutions or at least very flat minima. Much clearer and better-defined solutions may be obtained by adding a measure of non-void space to the objective function with little if any increase in structural compliance. We discuss various alternatives for cleaning up solutions and propose two efficient approaches which both introduce an auxiliary field to control non-void space: one approach based on a cut element based auxiliary field (hybrid approach) and another approach based on an auxiliary element based field (density approach). At the end, we demonstrate significant qualitative and quantitative improvements in variable thickness sheet and de-homogenization designs resulting from the proposed cleaning schemes.