In this paper, we focused on studying the number of limit cycles of a class of near-Hamiltonian systems, whose unperturbed system possesses a degenerate center. Using the first order Melnikov function, we obtained the lower bound of the maximum number of limit cycles in Poincaré bifurcation under certain conditions. In addition, we obtained the number of small-amplitude limit cycles that bifurcate from the degenerate center. We also provided two examples as applications.