What is the authenticity of the the Hadith that mentions the phrase: "There will come to you an extension (or aid) from the people of Yemen, so if you are able to have them seek forgiveness for you, then do so," and who is meant by it, what is its occasion, and who narrated it?

This hadith is about Uwais ibn Amir al-Qarani, and it was narrated by Imam Muslim in his Sahih. The story mentions that Umar ibn al-Khattab used to ask those coming from Yemen about Uwais ibn Amir until he found him. He asked him if Uwais ibn Amir was from Murad, then from Qaran, and if he had suffered from leprosy which had healed except for a dirham-sized spot, and if he had a mother to whom he was dutiful. Uwais affirmed all of this.

Umar said: I heard the Messenger of Allah, peace and blessings be upon him, say: "Uwais ibn Amir will come to you with the reinforcements from the people of Yemen, from Murad, then from Qaran. He had leprosy, and it healed except for a dirham-sized spot. He has a mother to whom he is dutiful. If he were to swear an oath by Allah, He would fulfill it for him. If you are able to have him seek forgiveness for you, then do so." So Umar asked him to seek forgiveness, and Uwais sought forgiveness for him.

The following year, a man from the nobles of Yemen performed Hajj and met Umar. Umar asked him about Uwais, and the man said that he had left him with a humble dwelling and few possessions. Umar repeated the aforementioned hadith. When the man went to Uwais and asked him to seek forgiveness, Uwais sought forgiveness for him after asking if he had met Umar. Then Uwais departed immediately after people became aware of him.