Using a delta-doped GaAs/AlGaAs heterostructure with a 10 nm spacer layer, we exploit the metastable nature of the DX centers at low temperatures to control electrostatically their net frozen charge density. The concentration of DX(-) centers at 77 K is only determined by the applied gate voltage V-c during the cool-down of the sample to 77 K, i.e., it is independent, on the time scale of noise experiments, of the subsequent change in the gate bias V-GS. The sheet carrier concentration n(s) of the two-dimensional electron gas is varied through the application of V-GS. Hall experiments performed at 77 K on gated Hall-bar structures show that V-c strongly affects both, the threshold voltage V-t and the exponent k, which enters into the observed power-law dependence of the Hall mobility mu on n(s). These dependencies were also studied directly on modulation doped field effect transistors from the analysis of their transfer characteristics I-DS-V-GS and g(me)-V-GS at low drain bias V-DS. The 1/f drain-current noise was investigated and, after subtraction of the noise arising from the series resistances, quantified by the extracted value of the channel-associated Hooge parameter alpha(ch). This parameter is found to depend on n, and exhibits the same power-law dependence as the reciprocal mobility 1/mu(n(s)). This striking correlation was established for various values of k and reveals screening effects on the 1/f noise. This correlation qualitatively supports the idea that the dominant mechanism of 1/f noise in modulation doped field effect transistors, at 77 K, is due to mobility fluctuations induced by screened fluctuations of Coulomb scattering, generated either by fluctuations of charge-state and/or motion of defects.