Sometimes called the Palace of Porphrogenitus. This was probably built in the later 13th or early 14th century and served as one of the imperial residences during the last two centuries of the Byzantine Empire. It was perhaps an annex of the nearby Palace of Blachernae. I quote from Strolling through Istanbul, the guidebook lags behind in mentioning that roof and floor have disappeared. The current building has been restored, and now functions, I was told by locals, as a hall for concerts and such.
It formerly housed a menagerie for a while, served as a brothel, then housed the Tefkur Saray pottery, the produce of which was of a quality below that of classical Iznik and showing European influences, but nice anyhow. Elsewhere I have an example or another.

It soon got into further decline, but now looks splendid. I found a site with a good 3D reconstruction of what it looked like in ancient times.
In 2023 I found it had become a fine museum.