We investigate the behavior of the power fluctuations of the focused annular and flat-topped beams when a realistic receiver possessing a finite sized aperture is employed in turbulent atmospheric optics links. Employing our previously derived formulation for the power scintillation index, the variations of the power scintillations and the receiver aperture averaging factor of the focused annular and flat-topped beams are scrutinized. Receiver aperture averaging factor is deduced from the ratio of power scintillation index detected by a finite sized aperture to that obtained by a point aperture. Influence of the receiver aperture radius, the propagation length, the structure constant, the inner and the outer beam sizes for an annular type incidence, flatness parameter for a flat-topped incidence and the focusing parameter for the multi-Gaussian beam in general, on the power scintillation and the receiver aperture averaging factor are studied. It is found that for the focused multi-Gaussian beams, the effect of the receiver aperture averaging factor increases as the aperture radius increases for larger link lengths. Additionally, for the annular incidences in turbulence, the effect of the receiver aperture averaging factor is stronger for larger inner beam source sizes. At a fixed receiver aperture radius, receiver aperture averaging becomes more effective when the structure constant becomes larger. When focused multi-Gaussian beams are compared to their collimated counterparts, it is seen that the receiver aperture averaging is more beneficial for the focused annular and focused flat-topped beams. At large link lengths, increase in the receiver aperture radius decreases the power scintillations.