There is a minhag some have to say special tefillos (some fast as well) on the day before ראש חודש (on some months it is omitted, e.g. when it is during a Yom Tov, e.g. חנוכה ), as part of Yom Kippur Katan, a custom from the Kabbalistic circle of Tzefas centuries ago.
One sees advertisements in publications at times, calling for observance of this minhag, on the recommendation of Maran Sar HaTorah, Rav Chaim Kanievsky zt"l.
What is interesting, however, is that Rav Chaim himself did not observe it. As reported in the fine Artscroll version of his great work ארחות יושר (p. 458), he followed (as in many other things) the practice of his uncle the Chazon Ish, who did not observe it.
א גוטען חודש