Clusters of galaxies are known to be turbulent environments, whether they are merging systems where turbulence is injected via the conversion of gravitational potential energy into the intracluster medium (ICM), or whether they are relaxed systems in which small-scale core sloshing is occurring within the potential well. In many such systems, diffuse radio sources associated with the ICM are found: radio haloes and mini-haloes. Abell 2142 is a rich cluster undergoing an extreme episode of core sloshing, which has given rise to four cold fronts and a complex multi-component radio halo. Recent work revealed that there are three primary components to the halo that spans a distance of up to around 2.4\,Mpc. The underlying physics of particle acceleration on these scales is poorly explored, and requires high-quality multi-frequency data with which to perform precision spectral investigation. We aim to perform such an investigation. We used new deep MeerKAT L-band (1283\,MHz) observations in conjunction with LOFAR HBA (143\,MHz) data as well as X-ray data from XMM-Newton and Chandra to study the spectrum of the halo and the connection between the thermal and non-thermal components of the ICM. We confirm the presence of the third halo component, detecting it for the first time at 1283\,MHz and confirming its ultra-steep spectrum nature, as we recovered an integrated spectrum of $ H3, \, total 0.10$. All halo components follow power-law spectra with increasingly steep spectra moving towards the cluster outskirts. We profiled the halo in three directions, finding evidence of asymmetry and spectral steepening along an axis perpendicular to the main axis of the cluster. Our investigation of the thermal non-thermal connection shows sub-linear correlations that are steeper at 1283\,MHz than 143\,MHz, and we find evidence of different connections in different components of the halo. In particular, we find both a moderate anti-correlation (H1, the core) and positive correlation (H2, the ridge) between the radio spectral index and X-ray temperature. Our results are broadly consistent with an interpretation of turbulent (re-)acceleration following an historic minor cluster merger scenario in which we must invoke some inhomogeneities. However, the anti-correlation between the radio spectral index and X-ray temperature in the cluster core is more challenging to explain; the presence of three cold fronts and a generally lower temperature may provide the foundations of an explanation, but detailed modelling is required to study this further.