How to say "What am I going to do with you" in Chinese?
Yes, there is a Chinese expression that almost perfectly matches the English one:

我该拿你怎么办

And it is used endearingly, or critically by superiors on inferiors. This also seems to match the English usage.

I don't know about its etymology; from its close match against English and its contemporaneity (Ngram Viewer shows it appeared in the 80's and began to spike from the 90's) I suspect it is highly influenced by the English expression.