We report on oxygen abundances determined from medium-resolution near-IR spectroscopy for a sample of 57 carbon-enhanced metal-poor (CEMP) stars selected from the Hamburg/ESO survey. The majority of our program stars exhibit oxygen-to-iron ratios in the range +0.5 < [O/Fe]< +2.0. The [O/Fe] values for this sample are statistically compared to available high-resolution estimates for known CEMP stars, as well as to high-resolution estimates for a set of carbon-normal metal-poor stars. Carbon, nitrogen, and oxygen abundance patterns for a sub-sample of these stars are compared to yield predictions for very metal-poor asymptotic giant-branch abundances in the recent literature. We find that the majority of our sample exhibit patterns that are consistent with previously studied CEMP stars having s-process-element enhancements, and thus have very likely been polluted by carbon- and oxygen-enhanced material transferred from a metal-poor asymptotic giant-branch companion.