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Context. Recent studies show that the radio power of quasars accreting at very high rates can reach surprisingly high values. These studies suggest that this radio emission could originate from star formation (SF); however, the lack of data does not rule out the presence of a jetted active galactic nucleus (AGN). Aims. We investigated the origin of the radio emission of a sample of 18 super-Eddington candidates, over a wide range of redshifts. These sources are expected to have an extreme radiative output per unit black hole mass, show high-velocity outflows, and are therefore thought to be prime mover of galactic evolution via radiative and mechanical feedback. Methods. We present new Karl G. Jansky Very Large Array (VLA) observations at the L, C, and X bands of these sources, which we combined with observations from the LOw-Frequency ARray (LOFAR) Two-meter Sky Survey (LoTSS) and the Very Large Array Sky Survey (VLASS). We also used optical and IR data to derive estimates of accretion and wind parameters, as well as star formation rates (SFRs), which we compared with those derived from the radio emission. Results. Based on radio variability, luminosity, morphology, radio spectral properties, radio versus IR estimates of SFR, and radio-to-mid IR flux ratio, we find that seven of our 18 targets likely have radio emission predominantly arising from SF, six from a combination of SF and AGN-related mechanisms, and only three from a core or jetted AGN origin. This result is consistent with previous studies and supports the prevalence of lower-power radio structures associated with star-forming activity rather than relativistic jets in the high Eddington ratio regime. In the same sample, however, the data suggest that three sources simultaneously exhibit super-Eddington accretion and relativistic ejections.