In the classical monogenic framework, the Szegö-Radon transform is abstractly defined as an orthogonal projection operator of a Hilbert module of monogenic functions onto a suitable closed submodule of plane waves with parameters in the Stiefel manifold $St(m,2)$. In this talk, we study a refinement of this transform in the hypermonogenic setting. Hypermonogenic functions form a subclass of monogenic functions arising in the study of a modified Dirac operator, which allows for weaker symmetries and also has a strong connection to the hyperbolic metric. In particular, we construct a projection operator from a module of hypermonogenic functions in $R^{p+q}$ onto a suitable submodule of plane waves parameterized now by a vector on the unit sphere of $R^q$. Moreover, we study the interaction of this Szegö-Radon transform with the generalized C-K extension operator. Finally, we develop a reconstruction (inversion) method for this transform.