Constraining the Southern Ocean carbon sink is vital for understanding global carbon cycling and climate. Monthly 1&deg;x1&deg; gridded products of surface seawater partial pressure of carbon dioxide (<em>p</em>CO₂) are routinely used to detect change in Southern Ocean carbon cycling. Despite Southern Ocean sampling being sparse and skewed to summer months, the uncertainty arising from incomplete sampling at the monthly 1&deg;x1&deg; scale is unquantified to date, and it remains unclear how this <em>p</em>CO₂ sampling error affects estimates of monthly gridded <em>p</em>CO₂ products and derived air-sea CO₂ fluxes. Here, we quantify the Southern Ocean <em>p</em>CO₂ sampling error using synthetic observations from an eddy-permitting ocean-sea ice-biogeochemistry configuration of the Energy Exascale Earth System Model. Using instantaneous snapshots of the model state at the times and locations of real-world ship and float observations, we reconstruct gridded monthly 1&deg;x1&deg; <em>p</em>CO₂ datasets over 2014&ndash;2023 and compare them with the simulated model <em>p</em>CO₂ output, isolating the error due to incomplete observational coverage at the monthly 1&deg;x1&deg; scale. We find that, on average, <em>p</em>CO₂ derived from synthetic observations is 13&ndash;14 <em>&mu;</em>atm lower than the modeled monthly mean, with individual grid cells exhibiting uncertainties up to an order of magnitude larger, primarily caused by undersampling during high-wind events in the Southern Ocean. Our results imply that the rate of oceanic CO₂ uptake derived from available <em>p</em>CO₂ observations is biased high. Since the <em>p</em>CO₂ uncertainty due to incomplete observational coverage exceeds other sources of uncertainty in gridded <em>p</em>CO₂ data products, e.g., measurement error, it represents a substantial, previously overlooked contribution that should be propagated through gap-filled <em>p</em>CO₂ data products and estimates of the Southern Ocean and global ocean carbon sink.