  Sources that leak Lyman-continuum (LyC) photons and lead to the reionisation
of the universe are intensely studied using multiple observing facilities.
Recently, the Low-redshift LyC Survey (LzLCS) has found the first large sample
of LyC emitting galaxies at low redshift ($z\sim 0.3$) with the Hubble Space
Telescope/Cosmic Origins Spectrograph. The LzLCS sample contains a robust
estimate of the LyC escape fraction ($f_\mathrm{esc}^\mathrm{LyC}$) for 66
galaxies spanning a wide range of $f_\mathrm{esc}^\mathrm{LyC}$. Here we, for
the first time, aim to study the radio continuum (RC) properties of LzLCS
sources and their dependence on $f_\mathrm{esc}^\mathrm{LyC}$. We present Karl
G. Jansky Very Large Array RC observations at C (4-8 GHz), S (2-4 GHz) and L
(1-2 GHz) bands for a sub-sample of the LzLCS sources. The radio spectral index
($\alpha^{\mathrm{3GHz}}_\mathrm{6GHz}$) spans a wide range from being flat (
$\geq -0.1$) to very steep ($\leq -1.0$). The strongest leakers in our sample
show flat $\alpha^{\mathrm{3GHz}}_\mathrm{6GHz}$, weak leakers have
$\alpha^{\mathrm{3GHz}}_\mathrm{6GHz}$ close to normal star-forming galaxies,
and non-leakers are characterized by steep
$\alpha^{\mathrm{3GHz}}_\mathrm{6GHz}$. We argue that a combination of young
ages, free-free absorption, and a flat cosmic-ray energy spectrum can
altogether lead to a flat $\alpha^{\mathrm{3GHz}}_\mathrm{6GHz}$ for strong
leakers. Non-leakers are characterized by steep spectra which can arise due to
break/cutoff at high frequencies. Such a cutoff in the spectrum can arise in a
single injection model of CRs characteristic of galaxies which have recently
stopped star formation. Such a relation between
$\alpha^{\mathrm{3GHz}}_\mathrm{6GHz}$ and $f_\mathrm{esc}^\mathrm{LyC}$ hints
at the interesting role of supernovae, CRs, and magnetic fields in facilitating
the escape ( and/or the lack) of LyC photons. (Abridged)
