  Many relaxed cool-core clusters host diffuse radio emission on scales of
hundreds of kiloparsecs: mini-haloes. However, the mechanism responsible for
generating them, as well as their connection with central active galactic
nuclei, is elusive and many questions related to their physical properties and
origins remain unanswered. This paper presents new radio observations of the
galaxy cluster Abell 1413 performed with MeerKAT (L-band; 872 to 1712 MHz) and
LOFAR HBA (120 to 168 MHz) as part of a statistical and homogeneous census of
mini-haloes. Abell 1413 is unique among mini-halo clusters as it is a
moderately-disturbed non-cool-core cluster. Our study reveals an asymmetric
mini-halo up to 584 kpc in size at 1283 MHz, twice as large as first reported
at similar frequencies. The spectral index is flatter than previously reported,
with an integrated value of $\alpha = -1.01 \pm 0.06$, shows significant
spatial variation, and a tentative radial steepening. We studied the
point-to-point X-ray/radio surface brightness correlation to investigate the
thermal/non-thermal connection: our results show a strong connection between
these components, with a super-linear slope of $b = 1.63 \pm 0.10$ at 1283 MHz
and $b = 1.20 \pm 0.12$ at 145 MHz. We also explore the X-ray surface
brightness/radio spectral index correlation, finding a slope of $b = 0.59 \pm
0.11$. Both investigations support the evidence of spectral steepening.
Finally, in the context of understanding the particle acceleration mechanism,
we present a simple theoretical model which demonstrates that hybrid scenarios
- secondary electrons (re-)accelerated by turbulence - reproduce a super-linear
correlation slope.
