  The minimal theory of quasidilaton massive gravity allows for a stable
self-accelerating de Sitter solution in a wide range of parameters. On the
other hand, in order for the theory to be compatible with local gravity tests,
the fifth force due to the quasidilaton scalar needs to be screened at local
scales. The present paper thus extends the theory by inclusion of a cubic
Horndeski term in a way that (i) respects the quasidilaton global symmetry,
that (ii) maintains the physical degrees of freedom in the theory being three,
that (iii) can accommodate the Vainshtein screening mechanism and that still
(iv) allows for a stable self-accelerating de Sitter solution. After adding the
Horndeski term (and a k-essence type nonlinear kinetic term as well) to the
precursor action, we switch to the Hamiltonian language and find a complete set
of independent constraints. We then construct the minimal theory with three
physical degrees of freedom by carefully adding a pair of constraints to the
total Hamiltonian of the precursor theory. Switching back to the Lagrangian
language, we study cosmological solutions and their stability in the minimal
theory. In particular, we show that a self-accelerating de Sitter solution is
stable for a wide range of parameters. Furthermore, as in the minimal theory of
massive gravity, the propagation speed of the massive gravitational waves in
the high momentum limit precisely agrees with the speed of light.
