A complete one-loop prediction for the single production of the neutral Higgs bosons in association with a photon in electron-positron collisions is presented in the framework the minimal supersymmetric standard model (MSSM), paying special attention to the individual contribution from each type of diagram. This process has no amplitude at tree level and is hence directly sensitive to one-loop impacts and the underlying dynamics of Higgs. To investigate the effect of the new physics, four different scenarios, which include a Higgs boson with mass and couplings consistent with those of the discovered Higgs boson and a considerable part of parameter space allowed by the bounds from the searches for additional Higgs bosons and sparticles, are chosen in the MSSM. The dependence of the cross section in both the standard model and MSSM on the c.m. energy is examined by considering the polarizations of the initial electron and positron beams. The effect of individual contributions from each type of one-loop diagram on the total cross section is also investigated in detail. Furthermore, the total cross sections of e(-)e(+) -> gamma h(0) as well as e(-)e(+) -> gamma A(0) are scanned over the plane m(A) - tan beta for each scenario. The full one-loop contributions are crucial for the analysis of beyond the standard model physics at a future electron-positron collider.