  We propose an Extended Hybrid High-Order scheme for the Poisson problem with
solution possessing weak singularities. Some general assumptions are stated on
the nature of this singularity and the remaining part of the solution. The
method is formulated by enriching the local polynomial spaces with appropriate
singular functions. Via a detailed error analysis, the method is shown to
converge optimally in both discrete and continuous energy norms. Some tests are
conducted in two dimensions for singularities arising from irregular geometries
in the domain. The numerical simulations illustrate the established error
estimates, and show the method to be a significant improvement over a standard
Hybrid High-Order method.
