  In this paper, we consider the Universe at the late stage of its evolution
and deep inside the cell of uniformity. At these scales, we can consider the
Universe to be filled with dust-like matter in the form of discretely
distributed galaxies, a $K$-essence scalar field, playing the role of dark
energy, and radiation as matter sources. We investigate such a Universe in the
mechanical approach. This means that the peculiar velocities of the
inhomogeneities (in the form of galaxies) as well as the fluctuations of the
other perfect fluids are non-relativistic. Such fluids are designated as
coupled because they are concentrated around the inhomogeneities. In the
present paper, we investigate the conditions under which the $K$-essence scalar
field with the most general form for its action can become coupled. We
investigate at the background level three particular examples of the
$K$-essence models: (i) the pure kinetic $K$-essence field, (ii) a $K$-essence
with a constant speed of sound and (iii) the $K$-essence model with the
Lagrangian $bX+cX^2-V(\phi)$. We demonstrate that if the $K$-essence is
coupled, all these $K$-essence models take the form of multicomponent perfect
fluids where one of the component is the cosmological constant. Therefore, they
can provide the late-time cosmic acceleration and be simultaneously compatible
with the mechanical approach.
