I have a connected graph $G=(V,E)$ in $n$ vertices. The edge weights are non-negative and form a metric space, thus for vertices $u,v,w \in V$ , such that $(u,v), (v,w), (w,u)\in E$ we have $r(u,w) \leq r(u,v)+r(v,w)$. We furthermore have the following condition: $\sum_{u\in V}R(u) \leq n$ where $R(u)$ is the average of the weights of the edges incident on $u$.

My question is, does there exist a minimum spanning tree, that has weight at most $Cn$ where $C$ is some universal constant? In place of a minimum weight spanning tree, a walk (sequence of connected vertices) such that the sum of weights of the walk is $Cn$ for some universal constant.