We report the first upper limits on the power spectrum of 21 cm fluctuations during the Epoch of Reionization and Cosmic Dawn from Phase II of the Hydrogen Epoch of Reionization Array (HERA) experiment. HERA Phase II constitutes several significant improvements in the signal chain compared to Phase I, most notably resulting in expanded frequency bandwidth, from 50–250 MHz. In these first upper limits, we investigate a small two-week subset of the available Phase II observations, with a focus on identifying new systematic characteristics of the instrument, and establishing an analysis pipeline to account for them. We report 2σ upper limits in eight spectral bands, spanning 5.6 ≤ z ≤ 24.4 that are consistent with thermal noise at the 2σ level for k ≳ 0.6–0.9 h Mpc−1 (band dependent). Our tightest limit during Cosmic Dawn (z > 12) is 1.13 × 106 mK2 at (k = 0.55 h Mpc−1, z = 16.78), and during the EoR (5.5 < z < 12), it is 1.78 × 103 mK2 at (k = 0.70 h Mpc−1, z = 7.05). We find that mutual coupling has become our dominant systematic, leaking foreground power that strongly contaminates the low-k modes, resulting in the loss of modes from k = 0.35 to 0.55 compared to Phase I data.