  We investigate the cosmological perturbation theory and calculate the
perturbation spectra in the warm $k$-inflation. Unlike the scalar perturbations
of $k$-inflation in the cold inflationary scenario, entropy perturbations
should be considered in our model. We study how the entropy perturbations
affect the evolution of curvature perturbation $\mathcal{R}$ on the comoving
hypersurfaces and find the dissipation coefficient $\Gamma$ plays a central
role in the entropy perturbations and the evolution equation of $\mathcal{R}$.
In the weak dissipation condition, entropy perturbations on the large scale can
be neglected such that the primordial spectrum of cosmological perturbations is
due only to adiabatic perturbations. Like general warm inflationary models,
density fluctuations also mainly originated from the thermal fluctuations of
the inflaton field rather than the vacuum fluctuations in our model. Finally,
we calculate the perturbation spectra and the tensor-to-scalar ratio, then find
the tensor-to-scalar ratio is smaller than that in the $k$-inflation or the
general potential-driven cold inflationary models.
