  We study the electronic spectral function of a metal in the vicinity of an
antiferromagnetic (AFM) quantum critical point, focusing on a situation where
the bare bandwidth of the spin fluctuations is significantly smaller than the
Fermi energy. In this limit, we identify a range of energies where the
fermionic quasiparticles near the "hot spots'' on the Fermi surface are
strongly scattered by the quantum critical fluctuations, whereas the damping of
the AFM fluctuations by the electrons is negligible. Within a one-loop
approximation, there is a parameter range where the $T=0$ spectral function at
the hot spots has a "precursor gap'' feature, with a local maximum at a finite
frequency. However, the ratio of the bare spin wave velocity to the Fermi
velocity required to obtain a precursor gap is probably too small to explain
experiments in the electron-doped cuprate superconductors (He et al., Proc.
Natl. Acad. Sci 116, 3449 (2019)). At lower frequencies, the Landau damping of
the AFM fluctuations becomes important, and the electronic spectral function
has the familiar ${\omega}^{-1/2}$ singularity. Our one-loop perturbative
results are supported by a numerical Monte Carlo simulation of electrons
coupled to an undamped, nearly-critical AFM mode.
