Within the framework of the linearized theory of electroelastic wave propagation, a model of a piezoelectric structure with a prestressed functionally graded coating made of piezoceramics of a trigonal system with a symmetry class of 3m is considered. The ferroelectric LiNbO₃ is used as the main material of the structure. The initial deformed state of the coating material is homogeneous, induced by the action of initial mechanical stresses and an external electrostatic field, the properties of the coating continuously change in thickness. By the example of the problem of the propagation of SH-waves from a remote source for structures with an inhomogeneous prestressed coating in the case of an electrically free and short-circuited surface, the influence of the nature and localization of the inhomogeneity of the coating on the features of SAW propagation is studied. The separate and combined effects of initial actions on changes in the physical properties of the structure, the transformation of the surface wave field, and the change in the SAW velocities in a wide frequency range is studied. The results obtained in this work are useful for understanding the dynamic processes in prestressed piezoelectric structures, in the optimization and design of new structures and devices on SAW with high performance characteristics.Within the framework of the linearized theory of electroelastic wave propagation, a model of a piezoelectric structure with a prestressed functionally graded coating made of piezoceramics of a trigonal system with a symmetry class of 3m is considered. The ferroelectric LiNbO₃ is used as the main material of the structure. The initial deformed state of the coating material is homogeneous, induced by the action of initial mechanical stresses and an external electrostatic field, the properties of the coating continuously change in thickness. By the example of the problem of the propagation of SH-waves from a remote source for structures with an inhomogeneous prestressed coating in the case of an electrically free and short-circuited surface, the influence of the nature and localization of the inhomogeneity of the coating on the features of SAW propagation is studied. The separate and combined effects of initial actions on changes in the physical properties of the structure, the transformation of the surface wave field, and the change in the SAW velocities in a wide frequency range is studied. The results obtained in this work are useful for understanding the dynamic processes in prestressed piezoelectric structures, in the optimization and design of new structures and devices on SAW with high performance characteristics.