In　this　paper,　we　study　a　class　of　piecewise　smooth　near-Hamiltonian　systems　with　piecewise　polynomial　perturbations.　We　first　give　the　expression　of　the　first　order　Melnikov　function,　and　then　estimate　the　number　of　limit　cycles　bifurcated　from　periodic　annuluses　by　Melnikov　function　method.　In　addition,　we　discuss　the　number　of　limit　cycles　that　can　appear　simultaneously　near　both　sides　of　a　generalized　homoclinic　or　generalized　double　homoclinic　loop.　　©　2023　World　Scientific　Publishing　Company.