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The axially deformed relativistic mean field theory with the effective interaction PK1 has been applied to investigate the contributions of time-odd fields to the ground-state binding energies of 66 odd-odd nuclei and 140 odd-A nuclei in O-Ca isotopes. The results show that the ground-state deformation and configuration of 5 nuclei have been changed with considering the time-odd fields. The odd-odd nuclei have two unpaired valence nucleons and corresponding contribution of time-odd fields to binding energies (0.15-0.90 MeV0.15-0.90 mEv) are larger than of the neighboring odd-A nuclei with only one unpaired valence nucleon (<0.4 MeV<0.4 MeV), and thus a staggering for this contribution appears in the isotones with odd number of neutrons. In addition, taking Al isotopes as an example, the contributions of time-odd fields to binding energies and corresponding configurations are studied in detail. Moreover, azimuthal components of neutron and proton currents in 22Al, 24Al and 26Al, and the differences between the contributions of time-odd fields to ground-state binding energies of mirror nuclei, as well as the odd-even staggering of binding energies, are also investigated. [ABSTRACT FROM AUTHOR]