The Independent Press Standards Organisation (Ipso) has upheld a complaint about the accuracy of the article, which was published in September.
The front page story, headlined “Court Jezter”, said Mr Corbyn had accepted Privy Council membership “so he can get his hands on £6.2 million” of Short Money.
It claimed a “constitutional crisis” could have been triggered if if the Opposition leader had declined to join, putting the money at risk.
But a ruling published on page two of the newspaper said: “Ipso’s Complaints Committee found that it was significantly misleading to claim that Labour’s access to the £6.2 million depended on whether Mr Corbyn was a member of the Privy Council.
“The two were not formally connected and the article did not make clear how a majority of the funding was in fact allocated. The Committee upheld the complaint as a breach of Clause 1.
“The newspaper failed to correct the significantly misleading coverage promptly and Ipso required The Sun to publish this adjudication.”