Neutrons produced by the 7Li(p,n)7Be reaction close to the reaction threshold are widely used to measure the cross section of s-process nucleosynthesis reactions. While experiments have been performed so far with Van de Graaff accelerators, future use of rf accelerators with higher intensities is planned to enable investigations on radioactive targets. Such accelerators have, however, poorer energy definition. We investigate in this work the effects of an energy-broadened proton beam on the spectrum of neutrons emitted in the thick-target 7Li(p,n) reaction at Ep = 1912 keV and on the experimental cross section of the 197Au(n,γ)198Au reaction in these conditions. Measurements were made with the bunched and chopped proton beam at the Van de Graaff facility of the Institute for Reference Materials and Measurements using the time-of-flight technique, both with a narrow-energy proton beam (standard deviation of the distribution σ ≈ 1.5 keV) and with a broad-energy proton beam (σ ≈ 20 keV) obtained with a gold foil degrader. Neutron spectra measured with the narrow-energy proton beam are consistent with those previously reported. The angle-integrated spectrum obtained with the broad-energy proton beam more closely resembles a Maxwellian-flux distribution. The measured neutron spectra agree well with Monte Carlo simulations that include experimentally measured cross sections, two-body kinematics and proton energy loss in the target. Gold activation studies were performed for both narrow-energy and broad-energy proton beams to check the impact of the ensuing spectra on this important reference standard for activation measurements.