Min-plus matrix multiplication is used in many problems operating on distances in graphs or solvable by dynamic programming. Assuming the APSP hypothesis, there is no subcubic-time algorithm for the min-plus product of two general $n\times n$ matrices, but structured matrices admit faster solutions. Planar graph algorithms often use Monge matrices, which have an $O(n^2)$-time min-plus multiplication procedure. Many results for sequence alignment problems, such as edit distance and longest increasing subsequence, apply simple unit-Monge matrices, whose min-plus product can be computed in $O(n\log n)$ time [Tiskin, SODA'10]. Russo [SPIRE'11] identified the core size $δ$ as the structural parameter behind the underlying matrix representation and showed an $O((n+δ)\log^3 n)$-time min-plus multiplication procedure for arbitrary Monge matrices. In this work, we prove a linear bound on the core size of the product matrix in terms of the core sizes of the input matrices and show how to solve the core-sparse Monge matrix multiplication problem in $O((n+δ)\log n)$ time, matching the complexity for simple unit-Monge matrices, where $δ= O(n)$. As witnessed by the recent work of Gorbachev and Kociumaka [STOC'25] for edit distance with integer weights, our generalization opens up the possibility of speed-ups for weighted sequence alignment problems. Furthermore, our multiplication algorithm can efficiently recover the witness for any entry of the output matrix. This allows us, for example, to preprocess an integer array of size $n$ in $\tilde{O}(n)$ time so that the longest increasing subsequence of any sub-array can be reconstructed in $\tilde{O}(\ell)$ time, where $\ell$ is the length of the reported subsequence. In comparison, Karthik C. S. and Rahul [arXiv, 2024] recently achieved $\tilde{O}(\ell+n^{1/2})$-time reporting after $\tilde{O}(n^{3/2})$-time preprocessing.