  We analyze data from the Hydrogen Epoch of Reionization Array. This is the
third in a series of papers on the closure phase delay-spectrum technique
designed to detect the HI 21cm emission from cosmic reionization. We present
the details of the data and models employed in the power spectral analysis, and
discuss limitations to the process. We compare images and visibility spectra
made with HERA data, to parallel quantities generated from sky models based on
the GLEAM survey, incorporating the HERA telescope model. We find reasonable
agreement between images made from HERA data, with those generated from the
models, down to the confusion level. For the visibility spectra, there is broad
agreement between model and data across the full band of $\sim 80$MHz. However,
models with only GLEAM sources do not reproduce a roughly sinusoidal spectral
structure at the tens of percent level seen in the observed visibility spectra
on scales $\sim 10$ MHz on 29 m baselines. We find that this structure is
likely due to diffuse Galactic emission, predominantly the Galactic plane,
filling the far sidelobes of the antenna primary beam. We show that our current
knowledge of the frequency dependence of the diffuse sky radio emission, and
the primary beam at large zenith angles, is inadequate to provide an accurate
reproduction of the diffuse structure in the models. We discuss implications
due to this missing structure in the models, including calibration, and in the
search for the HI 21cm signal, as well as possible mitigation techniques.
