Is it permissible to publish the Hadith that begins with, "When Adam descended, he circumambulated the House seven times...", knowing that its chain of narration is weak?

The aforementioned hadith (definition) is weak, and it was deemed weak by Al-Haythami and Al-Albani. This is because Al-Nadr ibn Tahir is very weak and steals hadith. However, Ibn Hajar mentioned that its various chains of transmission raise it to a level where it can be acted upon in the context of virtues of deeds (fada'il al-a'mal), such as supplication. In summary, there is no harm in supplicating with it privately for oneself, as it is good speech. However, it should not be attributed to the Prophet, peace and blessings be upon him, nor should people be commanded to supplicate with it.