  We present the discovery and the subsequent follow up of radio emission from
SDSS J130402.36$+$293840.6 (J1304+2938), the candidate host galaxy of the
gamma-ray burst GRB~200716C. The galaxy is detected in the RACS (0.89 GHz), the
NVSS, the Apertif imaging survey, and the FIRST (1.4 GHz), the VLASS (3 GHz),
and in public LOFAR (130-170 MHz), WISE (3.4-22 $\mu$m), and SDSS (z, i, r, g,
u filters) data. The luminosity inferred at 1.4 GHz is
(5.1$\pm0.2)\times10^{30}$ erg s$^{-1}$ Hz$^{-1}$. To characterise the emission
and distinguish between different components within the galaxy, we performed
dedicated, high-sensitivity and high-resolution observations with the European
VLBI Network + e-MERLIN at 1.6 and 5 GHz. We did not detect any emission from a
compact core, suggesting that the presence of a radio-loud AGN is unlikely, and
therefore we ascribe the emission observed in the public surveys to
star-forming regions within the galaxy. We confirm and refine the redshift
estimate, $z=0.341\pm0.004$, with a dedicated TNG spectroscopic observation.
Finally, we compiled a list of all the known hosts of GRB afterglows detected
in radio and computed the corresponding radio luminosity: if GRB~200716C
belongs to J1304+2938, this is the third most radio-luminous host of a GRB,
implying one of the highest star-formation rates (SFRs) currently known, namely
SFR$\sim$324$\pm$61 M$_{\odot}$ yr$^{-1}$. On the other hand, through the
analysis of the prompt emission light curve, recent works suggest that
GRB~200716C might be a short-duration GRB located beyond J1304$+$2938 and
gravitationally lensed by an intermediate-mass black hole (IMBH) hosted by the
galaxy. Neither the public data nor our VLBI observations can confirm or rule
out the presence of an IMBH acting as a (milli-)lens hosted by the galaxy, a
scenario still compatible with the set of radio observations presented in this
work.
