  We discuss the corrected thermodynamics and naked singularity structure of
the topological static spherically symmetric solution in
$\mathcal{F}(R,\mathcal{G})$ - gravity coupled with Born-Infeld - like
nonlinear electrodynamics. Solutions admitting black holes with constant
topological Euler density is analyzed in view of various thermodynamical
variables. The inclusion of logarithmic correction to the entropy is extended
to the other thermodynamical variables and the contribution of corrected
variables are displayed on various plots. The stability of black hole under the
effect of thermal variables is also studied. As a second scope of this study,
solutions admitting timelike naked singularity are probed with bosonic and
fermionic quantum wave packets to see if the singularity is quantum
mechanically regular or not. In this context, the evolution of these probes
remains well-defined if the corresponding spatial Hamiltonian operator is
essentially self-adjoint. Our calculations reveal that when the singularity is
probed with specific wave modes involving spin - $0$ and spin - $1/2$ quantum
wave packets, the corresponding wave operators turn out to be essentially
self-adjoint, which in turn implies unique well-defined time evolution.
