Find perfect matching faster than MCM, in graph that has perfect matching? Given an unweighted bipartite graph which has a perfect matching, is there an algorithm for finding a perfect matching in the graph that is faster than the best known algorithm for finding a maximum cardinality matching (MCM)? For example, is there some way to take advantage of the fact that the graph is known to have some perfect matching to help find one faster?

If yes, could you please help to point out some references? If it is impossible, could you briefly explain the reason? Comments and suggestions are also welcomed.
Hopcroft-Karp isn't necessarily optimal, and you can already find asymptotically faster algorithms in the relevant Wikipedia pages: https://en.wikipedia.org/wiki/Matching_(graph_theory)#In_unweighted_bipartite_graphs, https://en.wikipedia.org/wiki/Maximum_flow_problem#Solutions.  Note that you can use any network flow algorithm to find a perfect matching.

For instance, you can achieve $O(n^{2.373})$ running time using fast matrix multiplication, though this is only a theoretical result; in practice it's slower.  However, this is already enough to demonstrate that there's no way we're going to prove that you can't do better than Hopcroft-Karp (because you can).  For sparse graphs ($|E|=O(n)$), Madry's algorithm achieves $O(n^{1.43})$ running time.  Neither of these use/exploit the fact that the graph already contains a perfect matching.