  We investigate how feedback and environment shapes the X-ray scaling
relations of early-type galaxies (ETGs), especially at the low-mass end. We
select central-ETGs from the IllustrisTNG-100 box that have stellar masses
$\log_{10}(M_{\ast}/\mathrm{M_{\odot}})\in[10.7, 11.9]$. We derive mock X-ray
luminosity ($L_{\mathrm{X, 500}}$) and spectroscopic-like temperature
($T_{\mathrm{sl, 500}}$) of hot gas within $R_{500}$ of the ETG haloes using
the MOCK-X pipeline. The scaling between $L_{\mathrm{X, 500}}$ and the total
mass within 5 effective radii ($M_{5R_{\rm e}}$) agrees well with observed ETGs
from Chandra. IllustrisTNG reproduces the observed increase in scatter of
$L_{\mathrm{X, 500}}$ towards lower masses, and we find that ETGs with
$\log_{10} (M_{5R_{\rm e}}/\mathrm{M_{\odot}}) \leqslant 11.5$ with
above-average $L_{\mathrm{X, 500}}$ experienced systematically lower cumulative
kinetic AGN feedback energy historically (vice versa for below-average ETGs).
This leads to larger gas mass fractions and younger stellar populations with
stronger stellar feedback heating, concertedly resulting in the above-average
$L_{\mathrm{X, 500}}$. The $L_{\mathrm{X, 500}}$--$T_{\mathrm{sl, 500}}$
relation shows a similar slope to the observed ETGs but the simulation
systematically underestimates the gas temperature. Three outliers that lie far
below the $L_{\rm X}$--$T_{\rm sl}$ relation all interacted with larger galaxy
clusters recently and demonstrate clear features of environmental heating. We
propose that the distinct location of these backsplash ETGs in the $L_{\rm
X}$--$T_{\rm sl}$ plane could provide a new way of identifying backsplash
galaxies in future X-ray surveys.
