"with what purpose" is correct and usable in written English.
You can use it to introduce a question about someone's intention or motivation. For example, "With what purpose did she visit the library?".
Let's start with what purpose means.
"I don't know what I thought we were doing — with what purpose," Madley Croft says.
The basic questions that need to be answered are: What is evaluated where and with what purpose?
Three essential research questions are raised to be answered: (1) First, how and with what purpose are different aspects (migration, security, development, labour market and economy) linked within the Mobility Partnerships?
As we work toward finding gratitude in each triumph and tragedy, it is with whom and with what purpose we spend our time going forward that is telling.
If he's not screwing with batting stances, what purpose does he serve?
John McCain, in remarks to the Washington Post, summarized, with commendable honesty, what purpose the Gruber saga is playing: "This gives us ammunition to make fundamental changes to the law".
(Fitch has dual headquarters in New York and London, although its majority-owner is a French investor).Even if the EU could get away with this censorship, what purpose would it really serve?
MTV.com turned all this into an occasion for a think piece on Internet trolling, from which Tillman, after he and Spencer had finished goofing off, began to read aloud, with disdainful delight: "What purpose does trolling actually serve?
If paleontology is an adventure with a purpose, what purpose would all this history and heritage serve sitting in his lab?
