A balanced bipartite graph G is said to be 2p-Hamilton-biconnected if for any balanced subset W of size 2p of V(G), the subgraph induced by V(G)\W is Hamilton-biconnected. In this paper, we prove that “ Let G be a balanced bipartite graph of order 2n with minimum degree δ(G) ≥ k, where n ≥ 2k − p + 2 for two integers k ≥ p ≥ 0. If the number of edges e(G) > n(n − k + p − 1) + (k + 2)(k − p + 1), then G is 2p-Hamilton-biconnected except some exceptions.” Furthermore, this result is used to present two new spectral conditions for a graph to be 2p-Hamilton-biconnected. Moreover, the similar results are also presented for nearly balanced bipartite graphs