  We discuss cosmology based on a Cuscuta-Galileon gravity theory, which
preserves just two degrees of freedom. Although there exists no additional
degrees of freedom, introduction of a potential of a scalar field changes the
dynamics. The scalar field is completely determined by matter fields. Giving an
exponential potential as an example, we discuss the cosmological dynamics. The
gravitational "constant" $G_{\rm F}$ appeared in the effective Friedmann
equation becomes time dependent. We also present how to construct a potential
when we know the evolution of the Hubble parameter. When we assume the
$\Lambda$CDM cosmology for the background evolution, we find the potential
form.
  We then analyze the density perturbations, which equation is characterized
only by a change of the gravitational "constant"$G_{\rm eff}$, which also
becomes time dependent. From the observational constraints such as the
constraint from the big-bang nucleosynthesis and the constraint on
time-variation of gravitational constant, we restrict the parameters in our
models.
  Taking into account the time dependence of the gravitational constant in the
effective Friedmann equation, we may have a chance to explain the Hubble
tension problem.
