  We theoretically investigate a two-dimensional decorated honeycomb lattice
framework to realize a second-order topological magnon insulator (SOTMI) phase
featuring distinct corner-localized modes. Our study emphasizes the pivotal
role of spin-magnon mapping in characterizing bosonic topological properties,
which exhibit differences from their fermionic counterparts. We employ a
symmetry indicator topological invariant to identify and characterize this
SOTMI phase, particularly for systems respecting time-reversal and ${\sf{C}}_6$
rotational symmetry. Using a spin model defined on a honeycomb lattice
geometry, we demonstrate that introducing ``\textit{kekul\'e}'' type
distortions yields a topological phase. In contrast, ``\textit{anti-kekul\'e}''
distortions result in a non-topological magnonic phase. The presence of
kekul\'e distortions manifests in two distinct topologically protected bosonic
corner modes - an \textit{intrinsic} and a \textit{pseudo}, based on the
specific edge terminations. On the other hand, anti-kekul\'e distortions give
rise to \SW{\textit{Tamm/Shockley}} type bond-localized boundary modes, which
are non-topological and reliant on particular edge termination. We further
investigate the effects of random out-of-plane exchange anisotropy disorder on
the robustness of these bosonic corner modes. The distinction between SOTMIs
and their fermionic counterparts arises due to the system-specific magnonic
onsite energies, a crucial feature often overlooked in prior literature. Our
study unveils exciting prospects for engineering higher-order topological
phases in magnon systems and enhances our understanding of their unique
behavior within decorated honeycomb lattices.
