In about 70% of the population of relaxed, cool-core galaxy clusters, the brightest cluster galaxy (BCG) is radio loud, showing non-thermal radio jets and lobes ejected by the central active galactic nucleus (AGN). In recent years such relativistic plasma has been unambiguously shown to interact with the surrounding thermal intra-cluster medium (ICM) thanks to spectacular images where the lobe radio emission is observed to fill the cavities in the X-ray-emitting gas. This `radio feedback' phenomenon is widespread and is critical to understand the physics of the inner regions of galaxy clusters and the properties of the central BCG. At the same time, mechanically-powerful AGN are likely to drive turbulence in the central ICM which may also play a role for the origin of non-thermal emission on cluster-scales. Diffuse non-thermal emission has been observed in a number of cool-core clusters in the form of a radio mini-halo surrounding the radio-loud BCG on scales comparable to that of the cooling region. Large mini-halo samples are necessary to establish their origin and connection with the cluster thermal properties and dynamics, especially in light of future X-ray characterization of the cluster cores as it is expected by Athena-XIFU. We show that All-Sky reference survey at Band 2 with SKA1 at confusion limit (rms ~2 {\mu}Jy per beam) has the potential to detect up to ~620 mini-halos at redshift z<0.6, whereas Deep Tier reference surveys at Band 1/2 with SKA1 at sub-arcsec resolution (rms ~0.2 {\mu}Jy per beam) will allow a complete census of the radio-loud BCGs at any redshift down to a 1.4 GHz power of 10^{22} W/Hz. We further anticipate that SKA2 might detect up to ~1900 new mini-halos at redshift z<0.6 and characterize the radio-mode AGN feedback in every cluster and group up to redshift z ~1.7 (the highest-z where virialized clusters are currently detected).