  In this work we shall study $k$-inflation theories with non-minimal coupling
of the scalar field to gravity, in the presence of only a higher order kinetic
term of the form $\sim \mathrm{const}\times X^{\mu}$, with
$X=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$. The study will be focused
in the cases where a scalar potential is included or is absent, and the
evolution of the scalar field will be assumed to satisfy the slow-roll or the
constant-roll condition. In the case of the slow-roll models with scalar
potential, we shall calculate the slow-roll indices, and the corresponding
observational indices of the theory, and we demonstrate that the resulting
theory is compatible with the latest Planck data. The same results are obtained
in the constant-roll case, at least in the presence of a scalar potential. In
the case that models without potential are considered, the results are less
appealing since these are strongly model dependent, and at least for a
power-law choice of the non-minimal coupling, the theory is non-viable.
Finally, due to the fact that the scalar and tensor power spectra are conformal
invariant quantities, we argue that the Einstein frame counterpart of the
non-minimal $k$-inflation models with scalar potential, can be a viable theory,
due to the conformal invariance of the observational indices. The Einstein
frame theory is more involved and thus more difficult to work with it
analytically, so one implication of our work is that we provide evidence for
the viability of another class of $k$-inflation models.
