  The gray-body factor and the absorption cross section of the 4D ESTGB gravity
with a mode of nonlinear electrodynamics for the massless Dirac field are
studied in this paper. The magnetic charge value varies between
$-2^{(\frac{5}{3})}/3$ and $0$ as well as the ADM mass is set to $1$, which
corresponds to a non-extreme black hole. The gray-body factor is obtained using
the semi-analytic WKB method after solving the massless Dirac equation. When
the absolute value of magnetic charge is increasing, the gray-body factor
$\gamma(\omega)$ is decreasing. In addition, the partial absorption cross
section and the total absorption cross section are calculated by using the
partial wave method. We find that the maximum value of partial absorption cross
section decreases as $\kappa$ increases. And the existence of magnetic charge
causes the diminishing of the total absorption cross section. Finally, we find
that the absorption cross section of the Dirac field is more sensitive to
electric charge than magnetic charge by comparing the absorption cross section
of the Reissner-Nordstr$\rm\ddot{o}$m and ESTGB-NLED black holes.
