plate, to have a lot/enough on one's

have a lot on one's plate

Also, have too much on one's plate. Have a great deal (or too much) to cope with, as in What with the new baby and the new house, they have a lot on their plate, or I can't take that on now; I've got too much on my plate already. This expression transfers a loaded or overloaded dinner plate to other activities. [First half of 1900s]
See also: have, lot, on, plate
The American Heritage® Dictionary of Idioms by Christine Ammer.

plate, to have a lot/enough on one's

To have more than enough to worry about or cope with. This term originated in the 1920s and transferred the image of an overcrowded dinner plate to a very full agenda. R. Simons used it in Houseboat Killing (1959): “I’ll leave you at it. I’ve got plenty on my plate at the moment.” The synonymous my plate is full (or to have a full plate) date from the same period.
See also: enough, have, lot, on
The Dictionary of Clichés by Christine Ammer
See also:
  • a lot on (one's) plate
  • clean (one's) plate (up)
  • clean plate
  • accommodate (oneself) to (something)
  • accommodate oneself to
  • new
  • brand
  • brand spanking new
  • (brand) spanking new
  • acclimate