  We have used the Habitable Zone Planet Finder (HPF) to gather high
resolution, high signal-to-noise near-infrared spectra of 13 field red
horizontal-branch (RHB) stars, one open-cluster giant, and one very metal-poor
halo red giant. The HPF spectra cover the 0.81$-$1.28 \micron\ wavelength range
of the $zyJ$ bands, filling in the gap between the optical (0.4$-$1.0~\micron)
and infrared (1.5$-$2.4~\micron) spectra already available for the program
stars. We derive abundances of 17 species from LTE-based computations involving
equivalent widths and spectrum syntheses, and estimate abundance corrections
for the species that are most affected by departures from LTE in RHB stars.
Generally good agreement is found between HPF-based metallicities and abundance
ratios and those from the optical and infrared spectral regions. Light element
transitions dominate the HPF spectra of these red giants, and HPF data can be
used to derive abundances from species with poor or no representation in
optical spectra (\eg, \species{C}{i}, \species{P}{i}, \species{S}{i},
\species{K}{i}). Attention is drawn to the HPF abundances in two field
solar-metallicity RHB stars of special interest: one with an extreme carbon
isotope ratio, and one with a rare very large lithium content. The latter star
is unique in our sample by exhibiting very strong \species{He}{i} 10830~\AA\
absorption. The abundances of the open cluster giant concur with those derived
from other wavelength regions. Detections of \species{C}{i} and \species{S}{i}
in HD~122563 are reported, yielding the lowest metallicity determination of
[S/Fe] from more than one multiplet.
