  In this work, following the discrete de Rham (DDR) approach, we develop a
discrete counterpart of a two-dimensional de Rham complex with enhanced
regularity. The proposed construction supports general polygonal meshes and
arbitrary approximation orders. We establish exactness on a contractible domain
for both the versions of the complex with and without boundary conditions and,
for the former, prove a complete set of Poincar\'e-type inequalities. The
discrete complex is then used to derive a novel discretisation method for a
quad-rot problem which, unlike other schemes in the literature, does not
require the forcing term to be prepared. We carry out complete stability and
convergence analyses for the proposed scheme and provide numerical validation
of the results.
