Could a broken sleep cause headaches?

Possibly, especially: If your sleep is broken due to sleep apnea. Recommend sleep evaluation.

Sleep apnea: If your "broken sleep" is caused by sleep apnea, it means you are stopping breathing for at least 10 seconds ( and probably longer). That means your brain and your muscles are deprived of oxygen. That could be the cause of muscles spasms which results in pain. Another possibility is that you are grinding your teeth, which means instead of relaxing , your muscles are working too hard, causing pain.

Definitely, yes: Headache is a common disorder and represents a major health problem both in the US and abroad. Although physicians, particularly family practitioners, treat many patients with persistent headaches, there are also many headache sufferers who have never consulted a physician. Frequently, their headaches are not resolved and it is not unusual for a headache to accompany with sleep disorders.