  Context. Whereas X-ray clusters are extensively used for cosmology, their
idealistic modelling, through the hypotheses of spherical symmetry and
hydrostatic equilibrium, are more and more being questioned. Along these lines,
the soft X-ray emission detected in tens of clusters with ROSAT was found to be
higher than what expected from the idealistic hot gas modelling, pointing to
our incomplete understanding of these objects. Aims. Given that cluster
environments are at the interface between the hot intra-cluster medium (ICM),
warm circum-galactic medium (WCGM) and warm-hot intergalactic medium (WHIM), we
aim to explore the relative soft X-ray emission of different gas phases in
circum-cluster environments. Method. By using the most massive halos in
IllustrisTNG at z=0, we have predicted the hydrodynamical properties of the gas
from cluster centers to their outskirts (5 R200), and modelled their X-ray
radiation for various plasma phases. Results. First, we found that the radial
profile of temperature, density, metallicity and clumpiness of the ICM are in
good agreement with recent X-ray observations of clusters. Secondly, we have
developed a method to predict the radial profile of soft X-ray emission in
different bands, the column density of ions and the X-ray absorption lines (O
VIII, O VII, Ne IX, and Ne IX) of warm-hot gas inside and around clusters.
Conclusion. The warm gas (in the form of both WCGM and WHIM gas) is a strong
emitter in soft X-ray bands, and is qualitatively consistent with the
observational measurements. Our results suggest that the cluster soft excess is
induced by the thermal emission of warm gas in the circum-cluster environments.
