half a heart, with
With only moderate enthusiasm, as in After his transfer he worked with half a heart, looking forward to early retirement. [Mid-1800s] For an antonym, see with all one's heart.
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.