In　this　paper,　we　study　limit　cycle　bifurcations　for　planar　piecewise　smooth　near-Hamiltonian　systems　with　nth-order　polynomial　perturbation.　The　piecewise　smooth　linear　differential　systems　with　two　centers　formed　in　two　ways,　one　is　that　a　center-fold　point　at　the　origin,　the　other　is　a　center-fold　at　the　origin　and　another　unique　center　point　exists.　We　first　explore　the　expression　of　the　first　order　Melnikov　function.　Then　by　using　the　Melnikov　function　method,　we　give　estimations　of　the　number　of　limit　cycles　bifurcating　from　the　period　annulus.　For　the　latter　case,　the　simultaneous　occurrence　of　limit　cycles　near　both　sides　of　the　homoclinic　loop　is　partially　addressed.　©　2022,　The　Author(s),　under　exclusive　licence　to　Springer　Nature　Switzerland　AG.