What is the authenticity of the Hadith: "Whoever I am mentioned in front of, and he does not send prayers upon me, has been wretched," and what is our duty regarding the spread of fabricated Hadiths?

The hadith: "He who, when I am mentioned in his presence, does not invoke blessings upon me, has indeed become wretched," is weak because of Al-Fadl ibn Mubashshir, and scholars have unanimously agreed upon his weakness. It is superseded by an authentic hadith: "The miser is he in whose presence I am mentioned, and then he does not invoke blessings upon me." The duty regarding fabricated hadiths is to verify them, and if their weakness is confirmed, one must remain silent about them or explain their weakness to the people so they may beware of them.