Transport equations are investigated for a cylindrical plasma in the presence of electrostatic fluctuations. In a weakly turbulent regime, the transport matrices that relate the anomalous particle and heat fluxes and the parallel current to the thermodynamical forces are determined by employing drift-kinetic and gyrokinetic orderings for the electrons and ions. The calculation is based on the kinetic equations for the ensemble-averaged and fluctuating distribution functions. The crucial difference with previous works is the inclusion of an extra term in the drift-kinetic equation for the fluctuating electron distribution function. This extra term, which arises from the ensemble-averaged first-order (in a Larmor radius expansion) electron distribution function, leads to the Ware pinch components of the particle and heat fluxes and a correction to the Ohmic current. Furthermore, Shaing's ansatz, which was introduced in the synthetic theory of anomalous and neoclassical transport, is shown to be connected with this extra term in the context of a turbulent plasma, and the physical meaning and the validity of this ansatz are revealed. In drift-wave turbulence, the transport matrices, expressed in an implicit form by considering the frequency of fluctuations as a parameter, are rewritten in an explicit form by determining its frequency through the dispersion relation. The Onsager symmetry is shown to be broken for this explicit form of anomalous transport matrix.