  We examine the Chern Kondo insulator proposed in a square optical lattice
with staggered flux induced by $s$-$p$ orbital hybridization by revisiting its
realization and taking into account the magnetic effects for the Kondo phases.
The Ruderman-Kittel-Kasuya-Yoshida interaction is analyzed at the weak $s$-$p$
hybridization regime, with the anisotropic magnetic effects being discussed.
Furthermore, the paramagnetic and magnetic phases coexisting with Kondo
couplings are systematically investigated through the slave-boson theory, for
which the rich phases are obtained, including the antiferromagnetic, collinear
antiferromagnetic Kondo insulator, and Kondo metal phases. The magnetic orders
are shown to enhance the effective Kondo hybridization compared with the case
without taking into account magnetic effects, and exhibit different influences
on the bulk topology. In particular, the antiferromagnetic order always
enhances the topological phase by increasing bulk gap of the Chern Kondo
phases. The results show the rich topological and magnetic effects obtained in
the present Chern Kondo lattice model. We also investigate how to identify the
topology and strong correlation effects through measuring the Hall conductance
and double occupancy, which are achievable in ultracold atom experiments.
