  Quasinormal modes of black holes were previously calculated in a non-linear
electrodynamics and in the Gauss-Bonnet gravity theory. Here we take into
consideration both of the above factors and find quasinormal modes of a
(massive) scalar field in the background of a black hole in the
five-dimensional Einstein-Gauss-Bonnet gravity coupled to a non-linear
electrodynamics having Maxwellian weak-field limit. For the non-linear
electrodynamics we considered the high frequency (eikonal) regime of
oscillations analytically, while for the lower multipoles the higher order WKB
analysis with the help of Pad\'{e} approximants and the time domain integration
were used. We found that perturbations of a test scalar field violate the
inequality between the damping rate of the least damped mode and the Hawking
temperature, known as the Hod's proposal. This does not exclude the situation
in which gravitational spectrum may restore the Hod's inequality, so that only
the analysis of the full spectrum, including gravitational perturbations, will
show if the quasinormal modes we found here for the scalar field can be a
counterexample to the Hod's conjecture or not. We also revealed that in such a
system, which includes the higher curvature corrections and non-linear
electrodynamics, for perturbations of a massive scalar field there exists the
phenomenon of the arbitrary long lived quasinormal modes - quasiresonances.
