  We present improved photometric measurements for the host galaxies of 206
spectroscopically confirmed type Ia supernovae discovered by the Dark Energy
Survey Supernova Program (DES-SN) and used in the first DES-SN cosmological
analysis. Fitting spectral energy distributions to the $griz$ photometric
measurements of the DES-SN host galaxies, we derive stellar masses and
star-formation rates. For the DES-SN sample, when considering a 5D ($z$, $x_1$,
$c$, $\alpha$, $\beta$) bias correction, we find evidence of a Hubble residual
`mass step', where SNe Ia in high mass galaxies ($>10^{10} \textrm{M}_{\odot}$)
are intrinsically more luminous (after correction) than their low mass
counterparts by $\gamma=0.040\pm0.019$mag. This value is larger by $0.031$mag
than the value found in the first DES-SN cosmological analysis. This difference
is due to a combination of updated photometric measurements and improved star
formation histories and is not from host-galaxy misidentification. When using a
1D (redshift-only) bias correction the inferred mass step is larger, with
$\gamma=0.066\pm0.020$mag. The 1D-5D $\gamma$ difference for DES-SN is
$0.026\pm0.009$mag. We show that this difference is due to a strong correlation
between host galaxy stellar mass and the $x_1$ component of the 5D
distance-bias correction. To better understand this effect, we include an
intrinsic correlation between light-curve width and stellar mass in simulated
SN Ia samples. We show that a 5D fit recovers $\gamma$ with $-9$mmag bias
compared to a $+2$mmag bias for a 1D fit. This difference can explain part of
the discrepancy seen in the data. Improvements in modeling correlations between
galaxy properties and SN is necessary to determine the implications for
$\gamma$ and ensure unbiased precision estimates of the dark energy
equation-of-state as we enter the era of LSST.
