Can I control when the Auto schedules are allowed to run?

Yes — that's the "Active hours" card in the Scheduling section. Relay's Auto schedules run inside a window you set: pick the days of the week and a start and end time, and the repeating sync and push timers stay quiet outside it. Out of the box the window is Monday to Friday, 08:00–17:30 — working hours — so Relay isn't checking an untouched Supernote at three in the morning. Change the days or times to suit; an overnight window (say 20:00 to 02:00) works too, and belongs to the day it starts on.