  An algebra denoted $m\mathfrak{H}$ with three generators is introduced and
shown to admit embeddings of the Hahn algebra and the rational Hahn algebra. It
has a real version of the deformed Jordan plane as a subalgebra whose
connection with Hahn polynomials is established. Representation bases
corresponding to eigenvalue or generalized eigenvalue problems involving the
generators are considered. Overlaps between these bases are shown to be
bispectral orthogonal polynomials or biorthogonal rational functions thereby
providing a unified description of these functions based on $m\mathfrak{H}$.
Models in terms of differential and difference operators are used to identify
explicitly the underlying special functions as Hahn polynomials and rational
functions and to determine their characterizations. An embedding of
$m\mathfrak{H}$ in $\mathcal{U}(\mathfrak{sl}_2)$ is presented. A Pad\'e
approximation table for the binomial function is obtained as a by-product.
