In the article \(M_\alpha\)-Calderón-Zygmund operators are introduced and their \(L^p_\omega\) and \(H^p_\omega\) boundedness are considered. The class of \(M_\alpha\)-Calderón-Zygmund operators includes those of strongly singular Calderón-Zygmund operators and generalized Calderón-Zygmund operators with \(\delta\) standard kernel and the obtained results improve and extend those on latter classes.