A new renormalized statistical approach is proposed, that accounts for correlations between potential and magnetic fluctuations in a plasma. It is applicable for analysis of nonlinear model equations describing the cross-correlated states both for weak and strong turbulence. The influence of cross correlations on the spectral characteristics of potential and magnetic perturbations is investigated. It is shown that the non-zero cross-field correlations may influence drastically the spectra formation and affect the anomalous transport.