  In this paper we deal with a new class of Clifford algebra valued automorphic
forms on arithmetic subgroups of the Ahlfors-Vahlen group. The forms that we
consider are in the kernel of the operator $D \Delta^{k/2}$ for some even $k
\in {\mathbb{Z}}$. They will be called $k$-holomorphic Cliffordian automorphic
forms. $k$-holomorphic Cliffordian functions are well equipped with many
function theoretical tools. Furthermore, the real component functions have also
the property that they are solutions to the homogeneous and inhomogeneous
Weinstein equation. This function class includes the set of $k$-hypermonogenic
functions as a special subset. While we have not been able so far to propose a
construction for non-vanishing $k$-hypermonogenic cusp forms for $k \neq 0$, we
are able to do so within this larger set of functions. After having explained
their general relation to hyperbolic harmonic automorphic forms we turn to the
construction of Poincar\'e series. These provide us with non-trivial examples
of cusp forms within this function class. Then we establish a decomposition
theorem of the spaces of $k$-holomorphic Cliffordian automorphic forms in terms
of a direct orthogonal sum of the spaces of $k$-hypermonogenic Eisenstein
series and of $k$-holomorphic Cliffordian cusp forms.
