  Electron-phonon ($e$-ph) interactions are pervasive in condensed matter,
governing phenomena such as transport, superconductivity, charge-density waves,
polarons and metal-insulator transitions. First-principles approaches enable
accurate calculations of $e$-ph interactions in a wide range of solids.
However, they remain an open challenge in correlated electron systems (CES),
where density functional theory often fails to describe the ground state.
Therefore reliable $e$-ph calculations remain out of reach for many transition
metal oxides, high-temperature superconductors, Mott insulators, planetary
materials and multiferroics. Here we show first-principles calculations of
$e$-ph interactions in CES, using the framework of Hubbard-corrected density
functional theory (DFT+$U$ ) and its linear response extension (DFPT+$U$),
which can describe the electronic structure and lattice dynamics of many CES.
We showcase the accuracy of this approach for a prototypical Mott system, CoO,
carrying out a detailed investigation of its $e$-ph interactions and electron
spectral functions. While standard DFPT gives unphysically divergent and
short-ranged $e$-ph interactions, DFPT+$U$ is shown to remove the divergences
and properly account for the long-range Fr\"ohlich interaction, allowing us to
model polaron effects in a Mott insulator. Our work establishes a broadly
applicable and affordable approach for quantitative studies of e-ph
interactions in CES, a novel theoretical tool to interpret experiments in this
broad class of materials.
