  We consider modifications of general relativity characterized by a special
noncovariant constraint on metric coefficients, which effectively generates a
perfect-fluid type of matter stress tensor in Einstein equations. Such class of
modified gravity models includes recently suggested generalized unimodular
gravity (GUMG) theory and its simplest version -- unimodular gravity (UMG). We
make these gravity models covariant by introducing four Stueckelberg fields and
show that in the case of generalized unimodular gravity three out of these
fields dynamically decouple. This means that the covariant form of generalized
unimodular gravity is dynamically equivalent to k-essence theory with a
specific Lagrangian which can be reconstructed from the parameters of GUMG
theory. We provide the examples, where such reconstruction can be done
explicitly, and briefly discuss theories beyond GUMG, related to
self-gravitating media models. Also we compare GUMG k-inflation with cuscuton
models of dynamically inert k-essence field and discuss motivation for GUMG
coming from effective field theory.
