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In this paper we numerically explore the convergence properties of the circular restricted problem of three primary bodies, when an additional repulsive quasi-homogeneous Manev-type correction term is added to the effective potential. The Newton-Raphson iterative scheme is used for revealing the basins of convergence and their respective fractal basin boundaries on the complex plane. A thorough and systematic analysis is conducted in an attempt to determine the influence of the Manev parameter on the convergence properties of the system. Furthermore, the roots (numerical attractors) of the system and several other quantities (such as the required number of iterations, the percentages of the attracting regions and the basin entropy of the convergence diagrams) are monitored as a function of the Manev parameter, thus allowing us to extract useful correlations and conclusions. [ABSTRACT FROM AUTHOR]