  In interacting theories, hydrodynamics describes the universal behavior of
states close to local thermal equilibrium at late times and long distances in a
gradient expansion. In the hydrodynamic regime of metals, momentum relaxes
slowly with a rate $\Gamma$, which formally appears on the right-hand side of
the momentum dynamical equation and causes a Drude-like peak in the frequency
dependence of the thermoelectric conductivities. Here we study the structure
and determine the physical implications of momentum-relaxing gradient
corrections beyond Drude, \emph{i.e.} arising at subleading order in the
gradient expansion. We find that they effectively renormalize the weight of the
Drude pole in the thermoelectric conductivities, and contribute to the dc
conductivities at the same order as previously-known gradient corrections of
translation-invariant hydrodynamics. Turning on a magnetic field, extra
derivative corrections appear and renormalize the cyclotron frequency and the
Hall conductivity. This relaxed hydrodynamics provides a field-theoretic
explanation for previous results obtained using gauge-gravity duality. In
strongly-coupled metals where quasiparticles are short-lived and which may be
close to a hydrodynamic regime, the extra contributions we discuss are
essential to interpret experimentally measured magneto-thermoelectric
conductivities. Specializing to metals close to a Fermi liquid phase, the
effective mass measured either through the specific heat or the spectral weight
of the Drude-like peak are found to differ, as was indeed reported in overdoped
cuprate superconductors. More generally, we expect such terms to be present in
any hydrodynamic theory with approximate symmetries, which arise in many
physical systems.
