The Iraqi city of Sulaymaniyya became the capital of which short-lived unrecognized state in 1922?

That would be the Kingdom of Kurdistan, a rather short-lived entity born out of a series of revolts led by a certain Kurdish warlord in the person of Shaykh Mahmud Barzanji against the Iraqi authorities in the newly created British Mesopotamia.

The Kingdom of Kurdistan was actually just one of several attempts by the Kurds of what was then the British Mandate of Mesopotamia (largely modern day Iraq) to carve out an independent homeland for themselves.

The state was centred around the city of Sulaymaniyah, which was at the time the capital of Kurdistan.

The Kingdom of Kurdistan[2][3] was a short-lived unrecognised state proclaimed in the city of Sulaymaniyah following the collapse of the Ottoman Empire. Officially, the territory involved was under the jurisdiction of the British Mandate of Mesopotamia.

During the collapse of the Ottoman Empire, Kurds attempted to establish an independent state. On at least one occasion they succeeded and formed the Kingdom of Kurdistan, which lasted from September 1922 until July 1924.[4]

The Shaykh of the Qadiriyyah order of Sufis, the most influential personality in Southern Kurdistan,[5] was appointed governor of the former sanjak of Duhok, but rallied against the British and declared an independent Kurdistan in May 1919. He was defeated in June.

On 10 October 1921, a statement was issued in Suleymanyah, the capital of Kurdistan, to establish a Kurdish government. Sheikh Mahmud Barzanji declared himself as the King of the Kingdom of Kurdistan.[6][7][8]

https://en.m.wikipedia.org/wiki/Kingdom_of_Kurdistan