If you were taking the report to give to the director and he wasn't there, and you were leaving the report for him, you left it "with" the secretary. Only if you intended to give the reporter to the secretary and he/she wasn't there, would you leave it "for" the secretary.
You bought a gift for your friend, and you went to his house, but he wasn't there so you left it with his mother.
For kind of means because of. When you say for, like I left my report for the director, it (kind of) means I left my report because of the director. Using for will mean that the object - report - is intended to be given to the director.
Eg. You bought a gift for your friend For here will mean that the object - gift (it's the noun that precedes for) - is intended to be given to your friend (the noun that follows for).
When you say with, you only mean that you are leaving the report in the possession of the secretary for now, but do not intend to give the report to him ultimately.
Eg. but he wasn't there so you left it with his mother.
You only left the object - the gift - in the possession of his mother, but it is only temporary, since you want to give the gift to your friend, not his mother.