Solving a modified Travelling Salesman Problem(TSP)
As mentioned in the comments, you can compute APSP (All-Pairs Shortest Paths) at first and build the matrix $$D:=\left(d_{ij}\right)_{i,j\in[n]}$$, where $d_{ij}$ is the length of a shortest path from the vertex $i$ to the vertex $j$. By solving the usual travelling sales man on the graph $G'$ where $D$ is the adjacency matrix, you get a solution for your version of the problem. Try to see why this works as an exercise.

Your case is actually mentioned in the Wikipedia page under metrics.

Note that $D$ defines a metric and metric TSP admits a polynomial time 1.5-approximation. The same Wikipedia article contains more information about this algorithm under the section "Constructive heuristics".