  During the performance verification phase of the SRG/eROSITA telescope, the
eROSITA Final Equatorial-Depth Survey (eFEDS) has been carried out. It covers a
140 deg$^2$ field located at 126$^\circ <$ R.A. $< 146^\circ$ and -3$^\circ <$
Dec. $< +6^\circ$ with a nominal exposure over the field of 2.2 ks. 542
candidate clusters were detected in this field, down to a flux limit $F_X \sim
10^{-14}$ erg s$^{-1}$ cm$^{-2}$ in the 0.5-2 keV band. In order to understand
radio-mode feedback in galaxy clusters, we study the radio emission of
brightest cluster galaxies of eFEDS clusters, and we relate it to the X-ray
properties of the host cluster. Using LOFAR we identify 227 radio galaxies
hosted in the BCGs of the 542 galaxy clusters and groups detected in eFEDS. We
treat non-detections as radio upper limits. We analyse the properties of radio
galaxies, such as redshift and luminosity distribution, offset from the cluster
centre, largest linear size and radio power. We study their relation to the
intracluster medium of the host cluster. We perform statistical tests to deal
with upper limits on the radio luminosities. BCGs with radio-loud AGN are more
likely to lie close to the cluster centre than radio-quiet BCGs. There is a
clear relation between the cluster's X-ray luminosity and the radio power of
the BCG. Statistical tests indicate that this correlation is not produced by
selection effects in the radio band. We see no apparent link between largest
linear size of the radio galaxy and central density of the host cluster.
Converting the radio luminosity to kinetic luminosity, we find that radiative
losses of the intracluster medium are in an overall balance with the heating
provided by the central AGN. Finally, we tentatively classify our objects into
disturbed and relaxed, and we show that the link between the AGN and the ICM
apparently holds regardless of the dynamical state of the cluster.
