NP-completeness: sum of "some" paths in a spanning tree
You are asking for the minimum weight fundamental cycle basis (in an unweighted graph). I think the standard reference for its NP-hardness is:

Deo, Narsingh; Prabhu, G. M.; Krishnamoorthy, M. S. (1982), "Algorithms for generating fundamental cycles in a graph", ACM Transactions on Mathematical Software, 8 (1): 26–42, doi:10.1145/355984.355988.