The mechanical feedback from the central active galactic nuclei (AGNs) can be crucial for balancing the radiative cooling of the intracluster medium (ICM) at the cluster centre. We aim to understand the relationship between the power of AGN feedback and the cooling of gas in the centres of galaxy clusters by correlating the radio properties of the brightest cluster galaxies (BCGs) with the X-ray properties of their host clusters. We used the catalogues from the first SRG/eROSITA All-Sky Survey (eRASS1) along with radio observations from the Australian SKA Pathfinder (ASKAP). In total, we identified 134 radio sources associated with BCGs of the 151 eRASS1 clusters located in the PS1, PS2, and SWAG-X ASKAP fields. Non-detections were treated as upper limits. We correlated the radio properties of the BCGs (radio luminosity, largest linear size/LLS, and BCG offset from the cluster centre) with the integrated X-ray luminosity of the host clusters. We utilised the concentration parameter, $c_{R_{500}}$ , to categorise the clusters into cool cores (CCs) and non-cool cores (NCCs). By combining $c_{R_{500}}$ with the BCG offset, we assessed the dynamical states of the clusters in our sample. Furthermore, we analysed the correlation between radio mechanical power and X-ray luminosity within the CC subsample. We observe a potential positive trend between LLS and BCG offset, which may hint at an environmental influence on the morphology of central radio sources. We find a weak trend suggesting that more luminous central radio galaxies are found in clusters with higher X-ray luminosity. Additionally, there is a positive but highly scattered relationship between the mechanical luminosity of AGN jets and the X-ray cooling luminosity within the CC subsample. This finding is supported by bootstrap resampling and flux-flux analyses. The correlation observed in our CC subsample indicates that AGN feedback is ineffective in high-luminosity (high-mass) clusters. At a cooling luminosity of $L_{\mathrm{X},\,r} \lt \mathrm{R}_{\mathrm{cool}}\approx 5.50\times10^{43}\,\mathrm{erg\,s^{-1}}$ , on average, AGN feedback appears to contribute only about $13\%-22\%$ of the energy needed to offset the radiative losses in the ICM.
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