1. The tautological answer to the question "what is the centroid $x_C$ of $C$?" is "the centroid $x_C$ of $C$ is the centroid $x_C$ of $C$". It is hardly possible to give a better answer to this question without having the terms in which to express $x_C$ specified.  

2. The answer to your second question, "How is it related to $x_A+x_B$?" is "in no reasonable way". 

Indeed, for any natural $n\ge2$ and any vectors $a,b,c$ in $\Bbb R^n$
it is easy to find two compact convex subsets $A$ and $B$ of $\mathbb{R}^n$ with $x_A=a$, $x_B=b$, $x_{A+B}=x_C=c$. 

To do that start with $n=2$, let $A$ be the convex hull of the set $\{(-1,0),(1,0)\}$, and let $B$ be the convex hull of the set $\{(-2,0),(1,-1),(1,1)\}$. Then $x_A=x_B=(0,0)$ and 
$C=A+B$ is the convex hull of the set $\{(2,1),(2,-1),(-3,0),(0,1),(0,-1)\}$, so that $x_C=(1/7,0)\ne(0,0)=x_A+x_B$. 

Now -- by rescaling, embedding $\Bbb R^2$ into $\Bbb R^n$, rotating, and shifting -- for any natural $n\ge2$ and any vectors $a,b,c$ in $\Bbb R^n$ such that $c\ne a+b$ 
we get two compact convex subsets $A$ and $B$ of $\mathbb{R}^n$ with $x_A=a$, $x_B=b$, $x_C=c$. 

It is even easier, for any natural $n\ge2$ and any vectors $a,b$ in $\Bbb R^n$  
to get two compact convex subsets $A$ and $B$ of $\mathbb{R}^n$ with $x_A=a$, $x_B=b$, $x_C=a+b$.