A hydrodynamic model for steady-state, spherically symmetric winds driven by young stellar clusters with an exponential stellar density distribution is presented. Unlike in most previous calculations, the position of the singular point Rsp, which separates the inner subsonic zone from the outer supersonic flow, is not associated with the star cluster edge, but calculated self-consistently.When the radiative losses of energy are negligible, the transition from the subsonic to the supersonic flow occurs always at Rsp ≈ 4Rc, where Rc is the characteristic scale for the stellar density distribution, irrespective of other star cluster parameters. This is not the case in the catastrophic cooling regime, when the temperature drops abruptly at a short distance from the star cluster center, and the transition from the subsonic to the supersonic regime occurs at a much smaller distance from the star cluster center. The impact from the major star cluster parameters to the wind inner structure is thoroughly discussed. Particular attention is paid to the effects which radiative cooling provides to the flow. The results of the calculations for a set of input parameters, which lead to different hydrodynamic regimes, are presented and compared to the results from non-radiative one-dimensional numerical simulations and to those from calculations with a homogeneous stellar mass distribution.