The effects of the energy spread of secondary electrons are theoretically investigated for a dc-biased single-surface multipactor. In our previous publication [S. G. Jeon et al., Phys. Plasmas 16, 073101 (2009)], we obtained the conditions for the phase lock of an electron bunch, assuming zero velocity spread of the secondary electrons. In this work, we extended our previous theory to derive a quadratic map, by which the stability and bifurcation of the electron bunch can be systematically investigated. For the study of the energy spread of the secondary electrons, a randomized term was added to this map. The modified map then showed significant smearing-out of the bifurcated branches. The theoretical results were verified by particle-in-cell simulations, which showed good agreement in wide parameter ranges for both cases of monoenergetic and energy-spread secondary electrons.