To what extent is the hadith "Whoever desires poverty itself, let him sing at the time of prayer" authentic? And does this apply to the entire prayer time or only to the time of the call to prayer (adhan)? What should one who propagates it do if it is not authentic?

The mentioned Hadith has no found origin, and signs of fabrication are apparent in it. It contradicts reality, as there are wealthy individuals who indulge in musical instruments (ma'azif). The prohibition of musical instruments is established by authentic Hadiths, such as the suspended Hadith in Bukhari: "There will be among my Ummah people who will declare unlawful sexual intercourse, silk, wine, and musical instruments." There is no need to fabricate false Hadiths concerning this matter.

If you have published this Hadith, you must rectify what you have done by destroying any remaining copies, or by informing those to whom you distributed the Hadith that it is not authentic. This is an obligation upon you, along with repenting to Allah. Scholars have stated that a penitent person has fulfilled their duty, even if the effect of their action remains, similar to someone who retracts a widespread innovation after its followers have become numerous.

You must verify the authenticity of anything attributed to the Messenger of Allah (peace be upon him) before publishing it. Not everything heard is authentic. You should ask the people of knowledge specialized in Hadith, for Allah Almighty said: "So ask the people of the message if you do not know" (An-Nahl: 43). It is not permissible for non-specialists to transmit Hadiths without presenting them to the people of Hadith.