The goal of this paper is twofold: on the one hand, motivated by questions raised by Schenzel, we explore situations where the Hartshorne–Lichtenbaum Vanishing theorem for local cohomology fails, leading us to simpler expressions of certain local cohomology modules. As application, we give new expressions of the endomorphism ring of these modules. On the other hand, building upon previous work by Àlvarez Montaner, we exhibit the shape of Lyubeznik tables of the so-called partially sequentially Cohen–Macaulay rings as introduced by Sbarra and Strazzanti.