Contact. The extended ROentgen Survey with an Imaging Telescope Array (eROSITA) telescope on board the Spectrum-Roentgen-Gamma (SRG) mission has completed the first eROSITA All-Sky Survey (eRASS:1). It detected ∼10⁴ galaxy clusters in the western Galactic hemisphere. In the radio band, the Australian Square Kilometre Array Pathfinder (ASKAP) telescope completed its pilot 1 phase of the project Evolutionary Map of the Universe (EMU) with 220 000 sources in a 270 deg² field overlapping with eRASS:1. These two surveys are used to study radio-mode active galactic nuclei in clusters. Aims. In order to understand the efficiency of radio-mode feedback at the centers of galaxy clusters, we relate the radio properties of the brightest cluster galaxies to the X-ray properties of the host clusters. Methods. We identified the central radio sources in eRASS:1 clusters or calculated corresponding upper limits on the radio luminosity. Then, we derived relations between the X-ray properties of the clusters and the radio properties of the corresponding central radio source. Results. In total, we investigated a sample of 75 clusters. We find a statistically significant correlation between the X-ray luminosity of the cluster and the 944 MHz radio luminosity of the corresponding central radio galaxy. There is also a positive trend between the radio power and the largest linear size of the radio source. The density and the largest linear size are not correlated. We find that the kinetic luminosity of the radio jets in high-luminosity clusters with L_X > 10⁴³ erg s⁻¹ is no longer correlated with the X-ray luminosity, and we discuss various reasons. We find an anticorrelation between the central cooling time t_cool and the radio luminosity L_R, indicating a need for more powerful active galactic nuclei in clusters with short central cooling times.