Let $G$ be a connected hypergraph with even uniformity, which contains cut vertices. Then $G$ is the coalescence of two nontrivial connected sub-hypergraphs (called branches) at a cut vertex. Let $\mathcal{A}(G)$ be the adjacency tensor of $G$. The least H-eigenvalue of $\mathcal{A}(G)$ refers to the least real eigenvalue of $\mathcal{A}(G)$ associated with a real eigenvector. In this paper we obtain a perturbation result on the least H-eigenvalue of $\mathcal{A}(G)$ when a branch of $G$ attached at one vertex is relocated to another vertex, and characterize the unique hypergraph whose least H-eigenvalue attains the minimum among all hypergraphs in a certain class of hypergraphs which contain a fixed connected hypergraph.