  Einstein-Maxwell-Gauss-Bonnet-axion theory in $4$-dimensional spacetime is
investigated in this paper through a "Kaluza-Klein-like" process. Dual to
systems at finite temperature with background magnetic field on three
dimensions, the four-dimensional dyonic black hole solution coupled with higher
derivative terms is obtained. After the tensor-type perturbation is added, the
shear viscosity to entropy density ratio is calculated at high temperature and
low temperature separately. The behaviour of shear viscosity to entropy density
ratio of uncharged black holes is found to be similar with that in
$5$-dimensional spacetime, violating the Kovtun-Starinets-Son bound as well
when temperature becomes lower. In addition, the main feature of this ratio
remains almost unchanged in $4$ dimensions, which is characterised by
$(T/\Delta)^2$ at low temperature $T$, with $\Delta$ proportional to the
coefficient $\beta$ from scalar fields. The difficulty in causal analysis is
also discussed, which is mainly caused by the vanishing momentum term in
equations of motion.
