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 clusters 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.
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