The re-normalized quasi-linear theory of turbulence in uniform plasma, given by Mis-guich and Balescu, is generalized to the case of non-uniform plasma. In the weak-coupling and the weak non-uniformity approximation, the explicit expressions for the propagator and the average turbulent collision operator are obtained. Non-uniformity not only modifies the diffusion contribution, resonance broadening or frequency shift, Dupree damping and velocity-space slope effect of the average distribution function in these operators, but also producesnew differentio-exponential operators——the velocity differential operator in the propagatordue to non-uniformity of the time-space correlation function, and the space differential operator in the average turbulent collision operator due to non-uniformity of the average distribution function. Non-uniformity of the average distribution function prevents two free-stream propagators that are inverse to each other in the expression for the average turbulent collision operator in terms of the propagators from compensating for each other, and as a result the non-Markovian contribution in the propagator acted immediately on the average distribution function appears.
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