<p>It is known that second-order Studniarski derivatives can be used to compute tangents to the solution set of a generalized equation when standard (first-order) regularity conditions are absent, but relaxed (second-order) regularity conditions are fulfilled. This fact, roughly speaking, is only relevant in practice as long as the computation of the Studniarski derivatives itself does not incur any additional cost. However, by now the computation of these derivatives proved challenging. In this paper we explain how the second-order Studniarski derivative of the sum of a smooth single-valued and a generic set-valued mapping can be computed in terms of well-established first- and second-order objects from variational analysis. The key to these computations is a new verifiable condition that links first- and second-order information about the considered mappings. In addition, we address some tractable conditions guaranteeing relaxed regularity, and study applications to generalized equations with convex or polyhedral (set-valued) ingredients, including complementarity systems. Overall, our findings unify and improve a number of existing results on both the computation of second-order Studniarski derivatives and the computation of tangents to the solution set of a generalized equation under relaxed regularity conditions.</p>