Massive gravity, a gravity theory in which a graviton is given a non-zero mass, plays an important role in the late-time cosmology. Due to the inability to provide a non-trivial cosmological solution, a mass-varying massive gravity is introduced by allowing the graviton mass to change accordingly due to an extra scalar field. Unfortunately, the varying graviton mass shrinks in time and thus vanishes trivially at the moment of the late-time expansion. Here we present a new class of mass-varying massive gravity in which the graviton mass is now also a function of a kinetic term of the scalar field. Such a structure is quite remarkable since not only the graviton mass introduces an effective dark energy but there is another contribution which has a chance of being a dark matter candidate. Lastly, the cosmic evolution of this model is investigated through numerical calculations and it can be concluded that the cosmic evolution can alleviate the well-known cosmic coincidence problem.