Drug delivery by acoustic waves is a crucial technology for targeted therapy. Recently, a three-layered drug micro-particle was proposed and fabricated, the second shell of which greatly improves both the encapsulation of the drug and the flexibility in its release rate. In this work, the acoustic radiation force (ARF) of an acoustic focused Gaussian beam on a three-layered particle comprising an inner drug core (D), a middle layer of poly(lactide-co-glycolide) (PLGA), and an outer chitosan shell (CS) is investigated. A three-layered elastic shell (TES) mimics the D-PLGA-CS structure, and the acoustic scattering from and ARF of the D-PLGA-CS are studied using Mie theory. This paper focuses on how the geometry and acoustic parameters of the outer shell influence the ARF, finding that the Poisson's ratio of the outer shell affects the ARF more than does the density or Young's modulus. In addition, this paper finds that the choice of the inner drug has little effect on the ARF acting on the D-PLGA-CS particle. The present work may benefit the acoustic manipulation of both TESs and three-layered drugs.