Propagation of laser radiation with partial spatial coherence in a laterally bounded optically inhomogeneous extended gain medium is studied by use of quasi-optical equation for a transverse correlation function of field amplitude. The approach used allows for the radiation diffraction, refraction on a regular profile of dielectric constant, regular gain, scattering by fluctuations of dielectric constant and gain, scattering by random axial (hose-type) displacements. It is shown that fluctuations of the gain coefficient lead to additional regular gain of spontaneous emission, which dominates over the scattering by the fluctuations. Under the condition of small-scale hose-type fluctuations and low amplification within their longitudinal correlation length, fluctuations of the gain coefficient can hardly affect the intensity of amplified spontaneous emission, whereas their effect on coherence can be stronger.