  We analyze an analog of the $t$-$J$-$U$ model as applied to the description
of a single moir\'e flat band of twisted WSe$_2$ bilayer. To take into account
the correlation effects induced by a significant strength of the Coulomb
repulsion, we use the Gutzwiller approach and compare it with the results
obtained by the Hartree-Fock method. We discuss in detail the graduate
appearance of a two dome structure of the superconducting state in the phase
diagram by systematically increasing the Coulomb repulsion integral, $U$. The
two superconducting domes residing on both sides of a Mott insulating state can
be reproduced for a realistic parameter range in agreement with the available
experimental data. According to our analysis the paired state has a highly
unconventional character with a mixed $d+id$ (singlet) and $p-ip$ (triplet)
symmetry. Both components of the mixed paired state are of comparable
amplitudes. However, as shown here, a transition between pure singlet and pure
triplet pairing should be possible in the considered system by tuning the gate
voltage, which controls the magnitude of the valley-dependent spin-splitting in
the system.
