The valley-Hall effect provides topological protection to a broad class of defects in valley-Hall photonic
topological metamaterials. Unveiling precisely how such protection is achieved and its implications in practical
implementations is paramount to move from fundamental science to applications. To this end, we investigate
a honeycomb valley-Hall topological metamaterial and monitor the evolution of the topological valley-Hall
edge states and higher-order corner states under different perturbation δR. The evolutions of the edge states
of the armchair and zigzag interfaces are demonstrated, respectively. By adjusting the geometric parameters
and introducing disturbances to break the inversion symmetry, we achieve the edge states with different modes
including the conventional crossed edge state and the specific gapped edge state. It is found that the edge states
of topological valley kinking will gradually separate with the increase of δR, and finally a complete gap between
the edge states appears. The gap has rarely been reported previously in topological materials fabricated by
printed circuit board technology. In addition, the higher-order topological corner states can also be observed
in the proposed topological metamaterial. The higher-order topological phase is theoretically characterized by
nontrivial bulk polarization and the Wannier centers. Our results show that the corner state localization becomes
stronger with the increase of δR. It is expected that our results will provide a platform for the realization of
optical topological insulators.