  We report the most sensitive upper limits to date on the 21 cm epoch of
reionization power spectrum using 94 nights of observing with Phase I of the
Hydrogen Epoch of Reionization Array (HERA). Using similar analysis techniques
as in previously reported limits (HERA Collaboration 2022a), we find at 95%
confidence that $\Delta^2(k = 0.34$ $h$ Mpc$^{-1}$) $\leq 457$ mK$^2$ at $z =
7.9$ and that $\Delta^2 (k = 0.36$ $h$ Mpc$^{-1}) \leq 3,496$ mK$^2$ at $z =
10.4$, an improvement by a factor of 2.1 and 2.6 respectively. These limits are
mostly consistent with thermal noise over a wide range of $k$ after our data
quality cuts, despite performing a relatively conservative analysis designed to
minimize signal loss. Our results are validated with both statistical tests on
the data and end-to-end pipeline simulations. We also report updated
constraints on the astrophysics of reionization and the cosmic dawn. Using
multiple independent modeling and inference techniques previously employed by
HERA Collaboration (2022b), we find that the intergalactic medium must have
been heated above the adiabatic cooling limit at least as early as $z = 10.4$,
ruling out a broad set of so-called "cold reionization" scenarios. If this
heating is due to high-mass X-ray binaries during the cosmic dawn, as is
generally believed, our result's 99% credible interval excludes the local
relationship between soft X-ray luminosity and star formation and thus requires
heating driven by evolved low-metallicity stars.
