  We analyze the robustness of corner modes in topological photonic crystals,
taking a $C_6$-symmetric breathing honeycomb photonic crystal as an example.
First, we employ topological quantum chemistry and Wilson loop calculations to
demonstrate that the topological properties of the bulk crystal stem from an
obstructed atomic limit phase. We then characterize the topological corner
modes emerging within the gapped edge modes employing a semi-analytical model,
determining the appropriate real space topological invariants. For the first
time, we provide a detailed account of the effect of long-range interactions on
the topological modes in photonic crystals, and we quantify their robustness to
perturbations. We conclude that, while photonic long-range interactions
inevitably break chiral symmetry, the corner modes are protected by lattice
symmetries.
