We report the discovery of a Neptune-sized ( ${R}_{p}=3.87\pm 0.06{R}_{\oplus }$ ) transiting circumbinary planet, Kepler-1661 b, found in the Kepler photometry. The planet has a period of ∼175 days and its orbit precesses with a period of only 35 yr. The precession causes the alignment of the orbital planes to vary, and the planet is in a transiting configuration only ∼7% of the time as seen from Earth. As with several other Kepler circumbinary planets, Kepler-1661 b orbits close to the stability radius, and is near the (hot) edge of the habitable zone. The planet orbits a single-lined, grazing eclipsing binary, containing a 0.84 ${M}_{\odot }$ and 0.26 ${M}_{\odot }$ pair of stars in a mildly eccentric (e = 0.11), 28.2 day orbit. The system is fairly young, with an estimated age of ∼1─3 Gyr, and exhibits significant starspot modulations. The grazing eclipse configuration means the system is very sensitive to changes in the binary inclination, which manifests itself as a change in the eclipse depth. The starspots contaminate the eclipse photometry, but not in the usual way of inducing spurious eclipse timing variations. Rather, the starspots alter the normalization of the light curve, and hence the eclipse depths. This can lead to spurious eclipse depth variations, which are then incorrectly ascribed to binary orbital precession.
