cut (something) on (something)
1. To accidentally cut or gash a part of the body on something, typically a sharp object. We came back early because Jeff cut his foot on a shell at the beach.
2. To chop something into pieces while it is resting on a particular surface. I made some space so you can cut those vegetables on the counter.
cut something on something
1. to slice something on or against something, accidentally. I cut my finger on the knife. Maria cut her foot on the broken glass.
2. to slice something that is lying on something else. I cut the tomatoes on the cutting board your mother gave us.