  We investigate the Gorkov--Melik-Barkhudarov (GM) correction to superfluid
transition temperature in two-dimensional Fermi gases with Rashba spin-orbit
coupling (SOC) across the SOC-driven BCS-BEC crossover. In the calculation of
the induced interaction, we find that the spin-component mixing due to SOC can
induce both of the conventional screening and additional antiscreening
contributions that interplay significantly in the strong SOC regime. While the
GM correction generally lowers the estimate of transition temperature, it turns
out that at a fixed weak interaction, the correction effect exhibits a
crossover behavior where the ratio between the estimates without and with the
correction first decreases with SOC and then becomes insensitive to SOC when it
goes into the strong SOC regime. We demonstrate the applicability of the GM
correction by comparing the zero-temperature condensate fraction with the
recent quantum Monte Carlo results.
