  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).
