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.