  We introduce the spaces $A^p_{\alpha, \beta}(\Omega)$ of $L^p$-solutions to
the Vekua equation (generalized monogenic functions) $D
w=\alpha\overline{w}+\beta w$ in a bounded domain in $\mathbb{R}^n$, where
$D=\sum_{i=1}^n e_i \partial_i$ is the Moisil-Teodorescu operator, $\alpha$ and
$\beta$ are bounded functions on $\Omega$.
  The main result of this work consists of a Hodge decomposition of the $L^2$
solutions of the Vekua equation, from this orthogonal decomposition arises an
operator associated with the Vekua operator, which in turn factorizes certain
Schr\"odinger operators. Moreover, we provide an explicit expression of the
ortho-projection over $A^p_{\alpha, \beta}(\Omega)$ in terms of the well-known
ortho-projection of $L^2$ monogenic functions and an isomorphism operator.
Finally, we prove the existence of component-wise reproductive Vekua kernels
and the interrelationship with the Vekua projection in Bergman's sense.
