The objective of the present paper is to study the $W_{2}$-curvature tensor of the semi-symmetric non-metric connection in a Kenmotsu manifold. It is shown that if in $M^n$, $W_{2}^* = 0$, then $M^n$ is isometric to the hyperbolic space $H^n(-1)$, where $W_{2}^*$ is the $W_{2}$-curvature tensor of the semi-symmetric non-metric connection. Also, locally $W_{2}$-$\phi$-symmetric Kenmotsu manifold and $W_{2}$-$\phi$-recurrent Kenmotsu manifold with respect to the semi-symmetric non-metric connection have been studied.