  We study the energy-density dynamics at finite momentum of the
two-dimensional Kitaev spin-model on the honeycomb lattice. Due to
fractionalization of magnetic moments, the energy relaxation occurs through
mobile Majorana matter, coupled to a static $\mathbb{Z}_2$ gauge field. At
finite temperatures, the $\mathbb{Z}_2$ flux excitations act as an emergent
disorder, which strongly affects the energy dynamics. We show that sufficiently
far above the flux proliferation temperature, but not yet in the classical
regime, gauge disorder modifies the coherent low-temperature energy-density
dynamics into a form which is almost diffusive, with hydrodynamic momentum
scaling of a diffusion-kernel, which however remains retarded, primarily due to
the presence of two distinct relaxation channels of particle-hole and
particle-particle nature. Relations to thermal conductivity are clarified. Our
analysis is based on complementary calculations in the low-temperature
homogeneous gauge and a mean-field treatment of thermal gauge fluctuations,
valid at intermediate and high temperatures.
