We prove Kontsevich's homological mirror symmetry conjecture for certain mirror pairs arising from Batyrev--Borisov's dual reflexive Gorenstein cones' construction. In particular we prove HMS for all Greene--Plesser mirror pairs (i.e., Calabi--Yau hypersurfaces in quotients of weighted projective spaces). We also prove it for certain mirror Calabi--Yau complete intersections arising from Borisov's construction via dual nef partitions, and also for certain Calabi--Yau complete intersections which do not have a Calabi--Yau mirror, but instead are mirror to a Calabi--Yau subcategory of the derived category of a higher-dimensional Fano variety. The latter case encompasses Kuznetsov's $K3$ category of a cubic fourfold', which is mirror to an honest $K3$ surface; and also the analogous category for a quotient of a cubic sevenfold by an order-$3$ symmetry, which is mirror to a rigid Calabi--Yau threefold.