see under off and on.
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.
Examples from the Web for on again
The 24-year-old Buffalo native had a "date" that night, but she wouldn't tell her on-again, off-again boyfriend with whom.The Serial Killer Victim's Secret Life
Christine Pelisek, Roja Heydarpour
April 24, 2011
I moved back to the city and my two on-again off-again guys, steeling myself for hopeless fix-ups, just in case love might strike.The Summer of the Entenmann's Man
July 9, 2010
Cale, absent from this reunion, has had an on-again off-again feud with Reed and was barely mentioned.Velvet Underground Reunites!
December 9, 2009