  The connection between the thermal and non-thermal properties in galaxy
clusters hosting radio halos seems fairly well established. However, a
comprehensive analysis of such a connection has been made only for integrated
quantities (e.g. $L_X - P_{radio}$ relation). In recent years new-generation
radio telescopes have enabled the unprecedented possibility to study the
non-thermal properties of galaxy clusters on a spatially resolved basis. Here,
we perform a pilot study to investigate the mentioned properties on five
targets, by combining X-ray data from the CHEX-MATE project with the second
data release from the LOFAR Two meter Sky survey. We find a strong correlation
($r_s \sim 0.7$) with a slope less than unity between the radio and X-ray
surface brightness. We also report differences in the spatially resolved
properties of the radio emission in clusters which show different levels of
dynamical disturbance. In particular, less perturbed clusters (according to
X-ray parameters) show peaked radio profiles in the centre, with a flattening
in the outer regions, while the three dynamically disturbed clusters have
steeper profiles in the outer regions. We fit a model to the radio emission in
the context of turbulent re-acceleration with a constant ratio between thermal
and non-thermal particles energy density and a magnetic field profile linked to
the thermal gas density as $B(r) \propto n_{th}^{0.5}$. We found that this
simple model cannot reproduce the behaviour of the observed radio emission.
