The corresponding Herz-type Hardy spaces to new weighted Herz spaces $HK^{\beta ,p}_{\alpha ,q}$ associated with the Dunkl operator on $\mathbb R$ have been characterized by atomic decompositions. Later a new characterization of $HK^{\beta ,p}_{\alpha ,q}$ on the real line is introduced. This helped us in the work to characterize that the norms of the Herz-type Hardy spaces for the Dunkl Operator can be achieved by finite central atomic decomposition in some dense subspaces of them. Secondly, as an application we prove that a sublinear operator satisfying many conditions can be uniquely extended to a bounded operator in the Herz-type Hardy spaces for the Dunkl Operator.