  In recent years, the amplitude of matter fluctuations inferred from
low-redshift probes has been found to be generally lower than the value derived
from CMB observations in the $\Lambda$CDM model. This tension has been
exemplified by Sunyaev-Zel'dovich and X-ray cluster counts which, when using
their Planck standard cluster mass calibration, yield a value of $\sigma_8$ ,
appreciably lower than estimations based on the latest Planck CMB measurements.
In this work we examine whether non-minimal neutrino masses can alleviate this
tension substantially. We used the cluster X-ray temperature distribution
function derived from a flux-limited sample of local X-ray clusters, combined
with Planck CMB measurements. These datasets were compared to $\Lambda$CDM
predictions based on recent mass function, adapted to account for the effects
of massive neutrinos. Treating the clusters mass calibration as a free
parameter, we examined whether the data favours neutrino masses appreciably
higher than the minimal 0.06 eV value. Using Markov chain Monte Carlo methods,
we found no significant correlation between the mass calibration of clusters
and the sum of neutrino masses, meaning that massive neutrinos do not
noticeably alleviate the above-mentioned Planck CMB--clusters tension. The
addition of other datasets (BAO and Ly-$\alpha$) reinforces those conclusions.
As an alternative possible solution to the tension, we introduced a simple,
phenomenological modification of gravity by letting the growth index $\gamma$
vary as an additional free parameter. We find that the cluster mass calibration
is robustly correlated with the $\gamma$ parameter, insensitively to the
presence of massive neutrinos or/and additional data used. We conclude that the
standard Planck mass calibration of clusters, if consolidated, would represent
evidence for new physics beyond $\Lambda$CDM with massive neutrinos.
