In 2017, the California-Kepler Survey (CKS) published its first data release (DR1) of high-resolution optical spectra of 1305 planet hosts. Refined CKS planet radii revealed that small planets are bifurcated into two distinct populations, super-Earths (smaller than 1.5R{Earth}) and sub-Neptunes (between 2.0 and 4.0R{Earth}), with few planets in between (the "radius gap"). Several theoretical models of the radius gap predict variation with stellar mass, but testing these predictions is challenging with CKS DR1 due to its limited M{star} range of 0.8-1.4M{sun}. Here we present CKS DR2 with 411 additional spectra and derived properties focusing on stars of 0.5-0.8M{sun}. We found that the radius gap follows Rp{prop}Pm with m=-0.10{+/-}0.03, consistent with predictions of X-ray and ultraviolet- and core-powered mass-loss mechanisms. We found no evidence that m varies with M{star}. We observed a correlation between the average sub-Neptune size and M{star}. Over 0.5-1.4M{sun}, the average sub-Neptune grows from 2.1 to 2.6R{Earth}, following R_p_{prop}M_star_^{alpha}^ with {alpha}=0.25{+/-}0.03. In contrast, there is no detectable change for super-Earths. These M{star}-Rp trends suggest that protoplanetary disks can efficiently produce cores up to a threshold mass of Mc, which grows linearly with stellar mass according to Mc~10M{Earth}(M_star_/M{sun}). There is no significant correlation between sub-Neptune size and stellar metallicity (over -0.5 to +0.5dex), suggesting a weak relationship between planet envelope opacity and stellar metallicity. Finally, there is no significant variation in sub-Neptune size with stellar age (over 1-10Gyr), which suggests that the majority of envelope contraction concludes after ~1Gyr.