  Higher-order topological insulators (HOTIs) have attracted increasing
interest as a unique class of topological quantum materials. One distinct
property of HOTIs is the crystalline symmetry-imposed topological state at the
lower-dimensional outer boundary, e.g. the zero-dimensional (0D) corner state
of a 2D HOTI, used exclusively as a universal signature to identify
higher-order topology but yet with uncertainty. Strikingly, we discover the
existence of inner topological point states (TPS) in a 2D HOTI, as the embedded
"end" states of 1D first-order TI, as exemplified by those located at the
vacancies in a Kekule lattice. Significantly, we demonstrate that such inner
TPS can be unambiguously distinguished from the trivial point-defect states, by
their unique topology-endowed inter-TPS interaction and correlated magnetic
response in spectroscopy measurements, overcoming an outstanding experimental
challenge. Furthermore, based on first-principles calculations, we propose
{\gamma}-graphyne as a promising material to observe the higher-order TPS. Our
findings shed new light on our fundamental understanding of HOTIs, and also
open an avenue to experimentally distinguishing and tuning TPS in the interior
of a 2D sample for potential applications.
