A dielectric slab of thickness $t$ and dielectric constant $k$ is introduced between the plates of a parallel-plate capacitor (plate area $A$, separation $d$, $t < d$). The new capacitance is:

Inside the slab the field is reduced by $k$ over thickness $t$; outside it is unchanged over $d-t$. Net voltage for charge $Q$ is $V = (Q/A\varepsilon_0)[(d-t) + t/k]$. So $C = Q/V = A\varepsilon_0/[d - t(1 - 1/k)]$. The $kA\varepsilon_0/d$ form is only when the slab fills the gap ($t = d$).