Marie did indeed live in Kiev for a time. From my edition of Nicholas and Alexandra (pg. 358): "Early in November [1916], Nicholas, with Alexis, went to Kiev to inspect hospitals and to visit his mother, who was living away from Petrograd [...] In Kiev, Nicholas had thought to relax from the problems of war and government. Instead, in their first conversation Marie demanded that he dismiss Sturmer and push Rasputin away from the throne."
I don't know where she lived in Kiev or if indeed she moved there because of the Rasputin scandal, although I wouldn't be surprised if that was the motive.