  Let ${\bf u}$ be a nonzero linear functional acting on the space of
polynomials. Let $\mathbf{D}_{q,\omega}$ be a Hahn operator acting on the dual
space of polynomials. Suppose that there exist polynomials $\phi$ and $\psi$,
with $\mathrm{deg}\,\phi\leq2$ and $\mathrm{deg}\,\psi\leq1$, so that the
functional equation $$ \mathbf{D}_{q,\omega}(\phi {\bf u})=\psi{\bf u} $$
holds, where the involved operations are defined in a distributional sense. In
this note we state necessary and sufficient conditions, involving only the
coefficients of $\phi$ and $\psi$, such that ${\bf u}$ is regular, that is,
there exists a sequence of orthogonal polynomials with respect to ${\bf u}$. A
key step in the proof relies upon the fact that a distributional Rodrigues-type
formula holds without assuming that ${\bf u}$ is regular.
