  We present theoretically the thermal Hall effect of magnons in a
ferromagnetic lattice with a Kekul\'e-O coupling (KOC) modulation and a
Dzyaloshinskii-Moriya interaction (DMI). Through a strain-based mechanism for
inducing the KOC modulation, we identify four topological phases in terms of
the KOC parameter and DMI strength. We calculate the thermal magnon Hall
conductivity ${\kappa^{xy}}$ at low temperature in each of these phases. We
predict an unconventional conductivity due to a non-zero Berry curvature
emerging from band proximity effects in the topologically trivial phase. We
find sign changes of ${\kappa^{xy}}$ as a function of the model parameters,
associated with the local Berry curvature and occupation probability of the
bulk bands. Throughout, ${\kappa^{xy}}$ can be easily tuned with external
parameters such as the magnetic field and temperature.
