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. For the DES-SN sample, when considering a 5D (z, x1, c, α, β) bias correction, we find evidence of a Hubble residual ‘mass step’, where SNe Ia in high-mass galaxies (>1010M⊙) are intrinsically more luminous (after correction) than their low-mass counterparts by γ=0.040±0.019γ=0.040±0.019 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 γ=0.066±0.020γ=0.066±0.020 mag. The 1D−5D γ difference for DES-SN is 0.026±0.0090.026±0.009 mag. We show that this difference is due to a strong correlation between host galaxy stellar mass and the x1 component of the 5D distance-bias correction. Including an intrinsic correlation between the observed properties of SNe Ia, stretch and colour, and stellar mass in simulated SN Ia samples, we show that a 5D fit recovers γ 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 modelling correlations between galaxy properties and SN is necessary to ensure unbiased precision estimates of the dark energy equation of state as we enter the era of LSST.