  The TiSiCO-family monolayer $X_2Y$CO$_2$($X$=Ti, Zr, Hf; $Y$=Si, Ge) is a
two-dimensional second-order topological insulator with unique valley-layer
coupling in equilibrium condition. In this work, based on the four-band
tight-binding (TB) model of monolayer Ti$_2$SiCO$_2$ (ML-TiSiCO) and the
Floquet theory, we study the non-equilibrium properties of the ML-TiSiCO under
a periodic field of laser and a gate-electric field. We find the interaction
between the time-periodic polarized light and the electric field can lead to a
variety of intriguing topological phase transitions. By driving the system with
only circularly polarized light (CPL), a photoinduced topological phase
transition occurs from a second-order topological insulator to a Chern
insulator with a Chern number of $C=\pm$2, and the sign of the Chern number $C$
is determined by the chirality of the incident light. Further adding a
perpendicular electric field, we find that the ML-TiSiCO exhibits a rich phase
diagram, consisting of Chern insulators with different Chern numbers and
various topological semimetals. In contrast, since the linearly polarized light
(LPL) does not break time-reversal symmetry, the Chern number of the system
would not be changed under the irradiation of LPL. However, there still exist
many topological phases, including second-order topological insulator,
topological semi-Dirac, Dirac and valley-polarized Dirac semimetals under the
interaction between the LPL and the electric field. Our results not only
enhance the understanding of the fundamental properties of ML-TiSiCO but also
broaden the potential applications of such materials in optoelectronic devices.
