  We present deep near-infrared $K_{\rm s}$-band imaging for 35 of the 53
sources from the high-redshift ($z > 2$) radio galaxy candidate sample defined
in Broderick et al. (2022). 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_{\rm s} \approx 21.6$$-$$23.0$ mag
(2$''$ diameter apertures; AB). The remaining eight targets are not detected to
a median $3\sigma$ depth of $K_{\rm s} \approx 23.3$ 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 $L_{500\,{\rm MHz}} > 10^{27}$ W Hz$^{-1}$. By
plotting 150-MHz flux density versus $K_{\rm s}$-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 $3\sigma$ lower limit of $K_{\rm s} \gtrsim 23.8$ mag ($1.5''$ diameter
apertures); these five targets, along with a further source from Broderick et
al. (2022) with a deep non-detection ($K_{\rm s} \gtrsim 23.7$ mag; $3\sigma$;
$2''$ diameter aperture) in the Southern H-ATLAS Regions $K_{\rm s}$-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 $z
\gtrsim 4.2$ is a plausible scenario for our ultra-high-redshift candidates if
the stellar mass $M_{\rm *} \gtrsim 10^{10.5}\,{\rm M}_\odot$. [abridged]
