We present deep near-infrared $K_\textrm{s}$ -band imaging for 35 of the 53 sources from the high-redshift ( $z 2$ ) radio galaxy candidate sample defined in Broderick et al. (2022, PASA, 39, e061). These images were obtained using the High-Acuity Widefield K-band Imager (HAWK-I) on the Very Large Telescope. Host galaxies are detected for 27 of the sources, with $K_\textrm{s} ≈ 21.6$ -23.0 mag (2 $^''$ diameter apertures; AB). The remaining eight targets are not detected to a median $3\unicode{x03C3}$ depth of $K_\textrm{s} ≈ 23.3$ mag (2 $^''$ diameter apertures). We examine the radio and near-infrared flux densities of the 35 sources, comparing them to the known $z 3$ powerful radio galaxies with 500-MHz radio luminosities $L_{500 \textrm{MHz}} 10^{27}$ W Hz $^{-1}$ . By plotting 150-MHz flux density versus $K_\textrm{s}$ -band flux density, we find that, similar to the sources from the literature, these new targets have large radio to near-infrared flux density ratios, but extending the distribution to fainter flux densities. Five of the eight HAWK-I deep non-detections have a median $3\unicode{x03C3}$ lower limit of $K_\textrm{s} rsim 23.8$ mag (1 $.\!^''$ 5 diameter apertures); these five targets, along with a further source from Broderick et al. (2022, PASA, 39, e061) with a deep non-detection ( $K_\textrm{s} rsim 23.7$ mag; $3\unicode{x03C3}$ ; 2 $^''$ diameter aperture) in the Southern H-ATLAS Regions $K_\textrm{s}$ -band Survey, are considered candidates to be ultra-high-redshift ( $z 5$ ) radio galaxies. The extreme radio to near-infrared flux density ratios ( $ 10^5$ ) for these six sources are comparable to TN J0924 $-$ 2201, GLEAM J0856 $+$ 0223 and TGSS J1530 $+$ 1049, the three known powerful radio galaxies at $z 5$ . For a selection of galaxy templates with different stellar masses, we show that $z rsim 4.2$ is a plausible scenario for our ultra-high-redshift candidates if the stellar mass $M_\textrm{*} rsim 10^{10.5}$ M $_\odot$ . In general, the 35 targets studied have properties consistent with the previously known class of infrared-faint radio sources. We also discuss the prospects for finding more UHzRG candidates from wide and deep near-infrared surveys.