In calculations of heavy-atom molecules with the use of the shape-consistent relativistic effective core potential (RECP), only valence and some outer core shells are treated explicitly, with the shapes of spinors being smoothed in the atomic core regions and the small components of four-component spinors being excluded from calculations. Therefore, the computational time can be dramatically reduced. However, in the framework of the standard radially local RECP versions, any attempt to extend the space of the explicitly treated electrons beyond a certain limit does not improve the accuracy of the calculations. The errors caused by these (nodeless) RECPs can be as high as 2000 cm^-1 and more for the dissociation and transition energies even for the lowest lying excitations, which is unacceptable for many applications. Moreover, direct calculations of such properties as the electronic densities near heavy nuclei, the hyperfine structure, and the matrix elements of other operators singular on heavy nuclei are impossible as a result of the smoothing of the orbitals in the core regions. In this paper, ways to overcome these disadvantages of the RECP method are discussed. The developments of the RECP method suggested by the authors are studied in many high-precision calculations of atoms and of the TlH, HgH molecules. The technique of nonvariational restoration of the electronic structure in the cores of the heavy atoms in molecules is applied to calculation of the P,T-odd spin-rotational Hamiltonian parameters, including the weak interaction terms which break the symmetry with respect to the space inversion (P) and time-reversal invariance (T) in the PbF, HgF, BaF, and YbF molecules.