It is known that second-order (Studniarski) contingent 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 second-order contingent derivatives itself does not incur
any additional cost, but by now the computation of these derivatives proved chal-
lenging. In this paper we explain how the second-order contingent 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 analy-
sis. The key to these computations is a new verifiable condition that links first- and
second-order information about the considered mappings. In addition, we study some
tractable conditions guaranteeing relaxed regularity, and applications to generalized
equations with polyhedral (set-valued) ingredients, including complementarity sys-
tems. Overall, our findings unify and improve a number of existing results on both
the computation of second-order contingent derivatives and the computation of tan-
gents to the solution set of a generalized equation under relaxed regularity conditions.