  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 1.5 $R_E$) and sub-Neptunes (between
2.0 and 4.0 $R_E$), with few planets in between (the "Radius Gap.") Several
theoretical models of the Radius Gap predict variation with stellar mass, but
testing these predictions are challenging with CKS DR1 due to its limited
$M_\star$ range of 0.8-1.4 $M_\odot$. Here, we present CKS DR2 with 411
additional spectra and derived properties focusing on stars of 0.5-0.8
$M_\odot$. We found the Radius Gap follows $R_p \propto P^m$ with $m = -0.10
\pm 0.03$, consistent with predictions of XUV- 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 to 1.4
$M_\odot$, the average sub-Neptune grows from 2.1 to 2.6 $R_E$, following $R_p
\propto M_\star^\alpha$ with $\alpha = 0.25 \pm 0.03$. In contrast, there is no
detectable change for super-Earths. These $M_\star$-$R_p$ trends suggests that
protoplanetary disks can efficiently produce cores up to a threshold mass of
$M_c$, which grows linearly with stellar mass according to $M_c \approx 10
M_E~(M_\star / M_\odot)$. There is no significant correlation between
sub-Neptune size and stellar metallicity (over $-$0.5 to $+$0.5 dex) 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 to 10 Gyr), which suggests that the majority of envelope contraction
concludes after $\sim$1 Gyr.
