Add a matching to a Hamiltonian path to reduce the max distance between given pairs of vertices
This answer is incorrect.

Your friend is right. Your problem (as interpreted by Sasho) does not place any restriction on the cardinality of the matching $C$. Therefore, choose $C$ to be a matching between the pairs in $R$. Then for any positive integer $k$, the distance between every pair in $R$ is less than $k$.

Your problem becomes interesting if you force paths to contain edges from both the matching $C$ and the path $P$.