  We theoretically investigate the Floquet generation of second-order
topological superconducting (SOTSC) phase, hosting Majorana corner modes
(MCMs), considering a quantum spin Hall insulator (QSHI) with proximity induced
superconducting $s$-wave pairing in it. Our dynamical prescription consists of
the periodic kick in time-reversal symmetry breaking in-plane magnetic field
and four-fold rotational symmetry breaking mass term while these Floquet MCMs
are preserved by anti-unitary particle-hole symmetry. The first driving
protocol always leads to four zero energy MCMs (i.e. one Majorana per corner)
as a sign of a {\it{strong}} SOTSC phase. Interestingly, the second protocol
can result in a {\it{weak}} SOTSC phase, harbouring eight zero energy MCMs (two
Majorana states per corner), in addition to the {\it{strong}} SOTSC phase. We
characterize the topological nature of these phases by Floquet quadrupolar
moment and Floquet Wannier spectrum. We believe that relying on the recent
experimental advancement in the driven systems and proximity induced
superconductivity, our schemes may be possible to test in the future.
