  We investigate the Minimal Theory of Massive Gravity (MTMG) in the light of
different observational data sets which are in tension within the $\Lambda$CDM
cosmology. In particular, we analyze MTMG model, for the first time, with the
Planck-CMB data, and how these precise measurements affect the free parameters
of the theory. The MTMG model can affect the CMB power spectrum at large
angular scales and cause a suppression on the amplitude of the matter power
spectrum. We find that on adding Planck-CMB data, the graviton has a small,
positive, but non-zero mass at 68\% confidence level, and from this
perspective, we show that the tension between redshift space distortions
measurements and Planck-CMB data in the parametric space $S_8 - \Omega_m$ can
be resolved within the MTMG scenario. Through a robust and accurate analysis,
we find that the $H_0$ tension between the CMB and the local distance ladder
measurements still remains but can be reduced to $\sim3.5\sigma$ within the
MTMG theory. The MTMG is very well consistent with the CMB observations, and
undoubtedly, it can serve as a viable candidate amongst other modified gravity
theories.
