We investigate the form and evolution of the X-ray luminosity–temperature (LX–kT) relation of a sample of 114 galaxy clusters observed with Chandra at 0.1 < z < 1.3. The clusters were divided into subsamples based on their X-ray morphology or whether they host strong cool cores. We find that when the core regions are excluded, the most relaxed clusters (or those with the strongest cool cores) follow an LX–kT relation with a slope that agrees well with simple self-similar expectations. This is supported by an analysis of the gas density profiles of the systems, which shows self-similar behaviour of the gas profiles of the relaxed clusters outside the core regions. By comparing our data with clusters in the Representative XMM-Newton Cluster Structure Survey (REXCESS) sample, which extends to lower masses, we find evidence that the self-similar behaviour of even the most relaxed clusters breaks at around 3.5 keV. By contrast, the LX–kT slopes of the subsamples of unrelaxed systems (or those without strong cool cores) are significantly steeper than the self-similar model, with lower mass systems appearing less luminous and higher mass systems appearing more luminous than the self-similar relation. We argue that these results are consistent with a model of non-gravitational energy input in clusters that combines central heating with entropy enhancements from merger shocks. Such enhancements could extend the impact of central energy input to larger radii in unrelaxed clusters, as suggested by our data. We also examine the evolution of the LX–kT relation, and find that while the data appear inconsistent with simple self-similar evolution, the differences can be plausibly explained by selection bias, and thus we find no reason to rule out self-similar evolution. We show that the fraction of cool core clusters in our (non-representative) sample decreases at z > 0.5 and discuss the effect of this on measurements of the evolution in the LX–kT relation.