The logarithmic correction to Bekenshtein-Hawking entropy in the framework of 4D Einstein$-$Gauss$-$Bonnet gravity coupled with nonlinear electrodynamics is obtained. We explore the black hole solution with the spherically symmetric metric. The logarithmic term in the entropy has a structure similar to the entropy correction in the semi-classical Einstein equations which mimics the quantum correction to the area low. The energy emission rate of black holes and energy conditions are studied. Quasinormal modes of black holes are investigated. The gravitational lensing of light around BHs was investigated. We calculated the deflection angle for some model parameters.