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Considering first-order corrections to entropy and temperature of Reissner–Nordström–anti-de Sitter (RN–AdS) and Kerr–Newman–anti-de Sitter (KN–AdS) black holes, effects from the corrected entropy and temperature on thermodynamic potentials like internal energy, enthalpy, Helmholtz energy and Gibbs free energy are analyzed. It is shown that when the black hole is large, the effects on the thermodynamic potentials are negligible. However, when the black hole is small, the thermodynamic potential of these two black holes receive observable impacts from the corrections of the entropy and temperature. We also find that these two kinds of black holes behave differently on thermodynamic potentials, since the existence of the angular momentum for KN–AdS black holes. Besides, critical behaviours in phase transitions of these AdS black hole, specific heat and scalar curvature of equilibrium state space geometry are obtained. In both of the two cases, critical points are shown to be shifted. Our work extends the former researches on corrected thermodynamics and geometrothermodynamics for AdS black holes, as corrected temperature due to Hawking radiation is considered.