$\newcommand\conv{\operatorname{conv}}\newcommand\ext{\operatorname{ext}}\newcommand\p{\partial}$The answer is yes. Indeed, let $K:=\conv P$ (the convex hull of $P$), let $\p K$ be the boundary of $K$, and let $\ext K$ be the set of all extreme points of $K$. Then $$P\subseteq\conv P=K=\conv\ext K \\ =\conv(P\cap\ext K) \subseteq\conv(P\cap\p K)=\conv C.\quad\Box$$

$\newcommand\conv{\operatorname{conv}}\newcommand\ext{\operatorname{ext}}\newcommand\p{\partial}$The answer is yes. Indeed, let $K:=\conv P$ (the convex hull of $P$), let $\p K$ be the boundary of $K$, and let $\ext K$ be the set of all extreme points of $K$. Then $K=\conv\ext K$, $\ext K=P\cap\ext K$, $\ext K\subseteq\p K$, and $P\cap\p K=C$. So, $$P\subseteq\conv P=K=\conv\ext K \\ =\conv(P\cap\ext K) \subseteq\conv(P\cap\p K)=\conv C.\quad\Box$$