Radio-mode feedback associated with the active galactic nuclei (AGN) at the cores of galaxy clusters injects large amount of energy into the intracluster medium (ICM), offsetting radiative losses through X-ray emission. This mechanism prevents the ICM from rapidly cooling down and fueling extreme starburst activity as it accretes onto the central galaxies, and is therefore a key ingredient in the evolution of galaxy clusters. However, the influence and mode of feedback at high redshifts (z~1) remains largely unknown. Low-frequency sub-arcsecond resolution radio observations taken with the International LOFAR Telescope have demonstrated their ability to assist X-ray observations with constraining the energy output from the AGNs (or "cavity power") in galaxy clusters, thereby enabling research at higher redshifts than before. In this pilot project, we test this hybrid method on a high redshift (0.6<z<1.3) sample of 13 galaxy clusters for the first time with the aim of verifying the performance of this method at these redshifts and providing the first estimates of the cavity power associated with the central AGN for a sample of distant clusters. We were able to detect clear radio lobes in three out of thirteen galaxy clusters at redshifts 0.7<z<0.9, and use these detections in combination with ICM pressures surrounding the radio lobes obtained from standard profiles to calculate the corresponding cavity powers of the AGNs. By combining our results with the literature, the current data appear to suggest that the average cavity power peaked at a redshift of z~0.4 and slowly decreases toward higher redshifts. However, we require more and tighter constraints on the cavity volume and a better understanding of our observational systematics to confirm any deviation of the cavity power trend from a constant level.