Q. See question above: How important is really the recitation of Ledavid?
A. The recitation of Ledavid mentioned above is not written in Shulchan Aruch and the usual accompanying Poskim.
It is firstly mentioned in Sefer Shem Tov Koton and he quotes that that the one who recites it from Elul until Simchas Torah, is promised to live his days in goodness and win over those heavenly accusers one has. See also Nitei Gavriel (3: 2 n. 3).
Mishna Berura (581: 2) also mentions to recite Ledavid after saying the costumed Shir Shel Yom (Psalm of the day) mentioned above.
Rabbi A. Bartfeld as revised by Horav Yaakov Hirschman, Horav Dovid Pam, Horav Aharon Miller, Horav Chanoch Ehrentreu and Horav Kalman Ochs Shlit’a