  We obtain an exact spherically symmetric and magnetically charged black hole
solution in 4D Einstein-Gauss-Bonnet gravity coupled with nonlinear
electrodynamics with the Lagrangian ${\cal L}_{NED} = -{\cal
F}/(1+\sqrt{2\mid\beta{\cal F}\mid})$ (${\cal F}$ is the field strength
invariant). The thermodynamics of the black hole is studied within our model.
We calculate the Hawking temperature and the heat capacity of the black hole.
The phase transitions occur in the point where the Hawking temperature
possesses an extremum and the heat capacity diverges. It was shown that black
holes are thermodynamically stable in some range of event horizon radii when
the heat capacity and Hawking temperature are positive. The logarithmic
correction to the Bekenstein - Hawking BH entropy is obtained which mimics a
quantum correction. The dependence of the BH shadow radius on the model
parameters is studied. We also investigate the particle emission rate of the
BH. The stability of the BH is studied by analysing the quasinormal modes.
