  We study extreme mass-ratio binary systems in which a stellar mass compact
object spirals into a supermassive black hole surrounded by a scalar cloud.
Scalar clouds can form through superradiant instabilities of massive scalar
fields around spinning black holes and can also serve as a proxy for dark
matter halos. Our framework is fully relativistic and assumes that the impact
of the cloud on the geometry can be treated perturbatively. As a proof of
concept, here we consider a point particle in circular, equatorial motion
around a non-spinning black hole surrounded either by a spherically symmetric
or a dipolar non-axisymmetric scalar cloud, but the framework can in principle
be generalized to generic black hole spins and scalar cloud geometries. We
compute the leading-order power lost by the point particle due to scalar
radiation and show that, in some regimes, it can dominate over
gravitational-wave emission. We confirm the presence of striking signatures due
to the presence of a scalar cloud that had been predicted using Newtonian
approximations, such as resonances that can give rise to sinking and floating
orbits, as well as "sharp features" in the power lost by the particle at given
orbital radii. Finally, for a spherically symmetric scalar cloud, we also
compute the leading-order corrections to the black-hole geometry and to the
gravitational-wave energy flux, focusing only on axial metric perturbations for
the latter. We find that, for non-compact clouds, the corrections to the
(axial) gravitational-wave fluxes at high frequencies can be understood in
terms of a gravitational-redshift effect, in agreement with previous works.
