  We study the dissipative conductivity $\sigma_1$ of a dirty superconductor
with a finite Dynes parameter $\Gamma$ under a dc-biased weak time-dependent
field. The Usadel equation for the current-carrying state is solved to
calculate the pair potential, penetration depth, supercurrent density, and
quasiparticle spectrum. It is shown that, while the depairing current density
$j_d$ for $\Gamma=0$ is coincident with the Kupriyanov-Lukichev theory, a
finite $\Gamma$ decreases the superfluid density, resulting in a reduction of
$j_d$. The broadening of the peaks of the quasiparticle density of states
induced by a combination of a finite $\Gamma$ and a dc bias can reduce
$\sigma_1$ below that for the ideal dirty BCS superconductor with $\Gamma=0$,
while subgap states at Fermi level proportional to $\Gamma$ results in a
residual conductivity at $T\to 0$. We find the optimum combination of $\Gamma$
and the dc bias to minimize $\sigma_1$ by scanning all $\Gamma$ and all
currents up to $j_d$. By using the results, it is possible to improve $j_d$ and
reduce electromagnetic dissipation in various superconducting quantum devices.
