Which glands produce a hormone that is believed to affect the sleep cycle?

The pineal gland is responsible for the production of melatonin which is the hormone responsible for the regulation of sleeping and waking, or the sleep cycle.

Explanation:

The Pineal gland is a small pea-sized gland located just above the middle of the human brain. It is an endocrine (ductless) that secretes its hormone product, melatonin directly into the blood. it is inactive during the day as it is affected by light. However, in the dark and dimly-lit environments, the pineal gland is activated and thus begins to secrete melatonin into the blood resulting in feelings of drowsiness.

Melatonin production peaks at night. Melatonin receptors found on tissues specific for it are able to detect the peak in circulating melatonin at night and this signals to the body that it is night-time. Night-time levels of melatonin are at least 10-fold higher than daytime concentrations. This sleep-wake pattern over a 24-hour period is known as the circadian rythm.