Answer: First, let it be said that a ḥadīth should not be taken from a book of the sort you mentioned. One must return to the reliable ḥadīth collections—such as Ṣaḥīḥ al-Bukhārī, Ṣaḥīḥ Muslim, the Sunan works, and other well-known, trustworthy compilations. As for the report you cited, I have found no trace of it in the sources I surveyed, and it appears to have no basis whatsoever. The scholars who catalogued the virtues of Friday make no mention of it.

What is prescribed for Thursday night is to increase one’s invocation of blessings upon the Prophet ﷺ on that night and on Friday itself. At dawn on Friday it is recommended, in the Fajr prayer, to recite in the first rakʿah “Alif-Lām-Mīm al-Sajdah” (Sūrah 32) and in the second “Has there come upon man a period of time…?” (Sūrah 76). That is what is legislated for the eve of Friday. As for designating that night with a special prayer unlike the other nights, no authentic ḥadīth establishes it; the night is like any other. A Muslim should perform whatever night worship (tahajjud) he is able and conclude with witr. Performing the specific prayer you described is not supported by anything sound from the Prophet ﷺ.

As we have stated, if you wish to act upon a ḥadīth you must consult the well-known, reliable books of the Sunnah. Relying on an obscure or unknown book and transmitting ḥadīths from it will only lead you into error. The ḥadīth corpus contains fabricated reports—lies attributed to the Prophet ﷺ—as well as weak ones, and it is the works of the specialists that clarify these matters. Therefore, not every ḥadīth you find in a book may be acted upon until its source and authenticity have been verified. And Allah knows best.