The reaction γp → pπ⁰η has been studied with the CBELSA/TAPS detector at the electron stretcher accelerator ELSA in Bonn for incident photon energies from threshold up to 3.1 GeV. This paper has been motivated by the recently claimed observation of a narrow structure in the M_Nη invariant mass distribution at a mass of 1678MeV/c². The existence of this structure cannot be confirmed in the present work. Instead, for E_γ = 1400–1500 MeV and the cut M_pπ⁰ ≤ 1190MeV/c² a statistically significant structure in the M_pη invariant mass distribution near 1700 MeV/c² is observed with a width of Γ≈ 35 MeV/c² while the mass resolution is σ_res = 5 MeV/c². Increasing the incident photon energy from 1420 to 1540 MeV this structure shifts in mass from ≈ 1700 MeV/c² to ≈ 1725 MeV/c²; the width increases to about 50 MeV/c² and decreases thereafter. The cross section associated with this structure reaches a maximum of ≈ 100 nb around E_γ ≈ 1490 MeV (W ≈ 1920 MeV), which coincides with the pa₀ threshold. Three scenarios are discussed which might be the origin of this structure in the M_pη invariant mass distribution. The most likely interpretation is that it is due to a triangular singularity in the γp → pπ⁰η reaction