Let n∈N0, ψ be an analytic self-map on D and u be an analytic function on D. The single operator Du,ψn acting on various spaces of analytic functions has been a subject of investigation for many years. It is defined as (Du,ψnf)(z)=u(z)f(n)(ψ(z)),f∈H(D). However, the study of the operator Pu→,ψk, which represents a finite sum of these operators with varying orders, remains incomplete. The boundedness, compactness and essential norm of the operator Pu→,ψk on the Bloch space are investigated in this paper, and several characterizations for these properties are provided.