The key challenge in the observation of the redshifted 21-cm signal from cosmic reionization is its separation from the much brighter foreground emission. Such separation relies on the different spectral properties of the two components, although, in real life, the foreground intrinsic spectrum is often corrupted by the instrumental response, inducing systematic effects that can further jeopardize the measurement of the 21-cm signal. In this paper, we use Gaussian Process Regression to model both foreground emission and instrumental systematics in $\sim 2$ hours of data from the Hydrogen Epoch of Reionization Array. We find that a simple co-variance model with three components matches the data well, giving a residual power spectrum with white noise properties. These consist of an "intrinsic" and instrumentally corrupted component with a coherence-scale of 20 MHz and 2.4 MHz respectively (dominating the line of sight power spectrum over scales $k\parallel \le 0.2$ h cMpc$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$) and a baseline dependent periodic signal with a period of $\sim 1$ MHz (dominating over $k\parallel \sim 0.4-0.8$h cMpc$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$) which should be distinguishable from the 21-cm EoR signal whose typical coherence-scales is $\sim 0.8$ MHz.