The 12th entry for "leave" in OAAD:
-- to allow somebody to take care of something
leave somebody/something + adv./prep. You can leave the cooking to me.
She left her assistant in charge.
Leave it with me — I'm sure I can sort it out.
‘Where shall we eat?’ ‘I’ll leave it entirely (up) to you (= you can decide).’
They left me with all the clearing up.
leave somebody/something to do something I was left to cope on my own.
Does "leave sth to sb" have the same connotation as "leave sth with sb"?
We can see that the collocations of "leave" with preposition phrases are highly flexible. I reword these examples as:
You can leave me with the cooking.
You can leave the cooking with me.
Leave it to me — I'm sure I can sort it out.
Leave me with it — I'm sure I can sort it out.
They left all the clearing up with/to me.
Do they sound equally natural?