  In extended Kitaev models on the honeycomb lattice, off-diagonal interactions
(e.g. the $\Gamma, \Gamma^{'}$ terms) give rise to non-Kitaev quantum spin
liquid (QSL) and several magnetically ordered phases. In the present work, we
dope holes to the system and study the resultant $t$-$K$-$\Gamma$-$\Gamma^{'}$
model by mean field theory. The interplay between the charge and spin degrees
of freedom results in a rich phase diagram. Similar to doped cuprates,
superconductors, pseudogap phases, fermi liquid, strange metal and paramagnetic
phase are generated. What is different is that, more than one superconducting
phases (including a topological one) and more than one pseudogap phases are
obtained no matter what the original spin state is. The Chern number of the
topological superconductor is either $\nu=\pm2$ or $\nu=\pm1$, depending on the
ratio $\Gamma/ |K|$ in the spin channel. We further find that an intermediate
in-plane magnetic field can slightly enlarge the size of the topological
superconducting phase.
