  Solitons formed through the one-dimensional mass-kink mechanism on the edges
of two-dimensional systems with non-trivial topology play an important role in
the emergence of higher-order (HO) topological phases. In this connection, the
existing work in time-reversal symmetric systems has focused on gapping the
edge Dirac cones in the presence of particle-hole symmetry, which is not suited
to the common spin-Chern insulators. Here, we address the emergence of edge
solitons in spin-Chern number of $2$ insulators, in which the edge Dirac cones
are gapped by perturbations preserving time-reversal symmetry but breaking
spin-$U(1)$ symmetry. Through the mass-kink mechanism, we thus explain the
appearance of pairwise corner modes and predict the emergence of extra charges
around the corners. By tracing the evolution of the mass term along the edge,
we demonstrate that the in-gap corner modes and the associated extra charges
can be generated through the $S_z$-mixing spin-orbit coupling via the mass-kink
mechanism. We thus provide strong evidence that an even spin-Chern-number
insulator is an HO topological insulator with protected corner charges.
