The two-dimensional (2D) hybrid structures of boron nitride (BN) and graphene with properties superior to the individuals are long desired. In this work, we demonstrate theoretically that this goal can be reached in a new graphene-like borocarbonitride (g-BC6N) whose domain has been synthesized in recent experiments. It has a direct band gap of 1.833 eV and a high carrier mobility comparable to that of black phosphorene. The inversion symmetry breaking in g-BC6N leads to a pair of inequivalent valleys with opposite Berry curvatures in the vicinities of the vertices (K and K′) of the Brillouin zone. The coexistence of valley-selective circular dichroism and high carrier mobility in g-BNC6 is beneficial to realize the valley Hall effect. We also propose a tight-binding (TB) model to describe the intrinsic features of this type of lattice, revealing a new class of 2D valleytronic materials.