The object of the present paper is to study a special type of semi-symmetric non-metric $\phi$-connection on a Kenmotsu manifold. It is shown that if the curvature tensor of Kenmotsu manifolds admitting a special type of semi-symmetric non-metric $\phi$-connection $\bar{\nabla}$ vanishes, then the Kenmotsu manifold is locally isometric to the hyperbolic space $H^n(-1)$. Beside these, we consider Weyl conformal curvature tensor of a Kenmotsu manifold with respect to the semi-symmetric non-metric $\phi$-connection. Among other results, we prove that the Weyl conformal curvature tensor with respect to the Levi-Civita connection and the semi-symmetric non-metric $\phi$-connection are equivalent. Moreover, we deal with $\phi$-Weyl semi-symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric $\phi$-connection. Finally, an illustrative example is given to verify our result.