  Both mixtures of atomic Bose-Einstein condensates and systems with atoms
trapped in optical lattices have been intensely explored theoretically, mainly
due to the exceptional developments on the experimental side. We investigate
the properties of ultracold atomic impurities (bosons) immersed in a vortex
lattice of a second Bose-condensed species. In contrast to the static
optical-lattice configuration, the vortex lattice presents intrinsic dynamics
given by its Tkachenko modes. These excitations induce additional correlations
between the impurities, which consist in a long-range attractive potential and
in a density-dependent hopping, described here in the framework of an extended
Bose-Hubbard model. We compute the quantum phase diagram of the impurity
species through a Gutzwiller ansatz and through the mean-field approach, and
separately identify the effects of the two additional terms, i.e., the shift
and the deformation of the Mott insulator lobes. The long-range attraction, in
particular, induces the existence of a triple point in the phase diagram, in
agreement with previous quantum Monte Carlo calculations [Chaviguri \emph{et
al.}, Phys. Rev. A \textbf{95}, 053639 (2017)].
