Question:  Another ship was supposed to sail with the Mayflower but had to turn back. What was the name of the other ship?

The answer is: The Speedwell.

Read More About The Speedwell Here

The Leiden Separatists bought a small ship, the Speedwell, in Holland. They embarked from Delftshaven on July 22, 1620. 
They sailed to Southampton, England to meet the Mayflower, which had been chartered by their English investors. There, other Separatists and additional colonists joined them.

On August 15, the Mayflower and Speedwell set sail for America. The Speedwell leaked so badly that both ships turned back to England, putting in first at Dartmouth and then at Plymouth. Finally, on September 16, 1620, the Mayflower set sail, alone, for America.