A method is described for the calculation of light-scattering properties of randomly oriented, axially symmetric spheroid particles with inclusion, in the framework of the T-matrix theory. Both the core and the shell can be axially symmetric nonspherical particles, and the inner and outer layers can be concentric or eccentric. Based on the model of spheroid particle with inclusion, light scattering characteristics of water aerosols with an absorptive core are calculated. The effects of core size, shape and position on the light backscattering characteristics are analyzed. It is suggested that light scattering properties in the backscattering region are sensitive to the shape, size, position of the core and the shape of the shell.