Are there poskim who permit cutting nails on Rosh Chodesh that falls out on erev Shabbos? May one be lenient with this in a pressing situation?
a gutten chodesh/ a gutten Shabboos/ a gutten Shavuous
It is permitted to cut nails in honor of Shabbos, if this is something that you are used to doing. Even if you don’t cut your nails every week, but only when they are a little long, in honor of Shabbos, it is permitted.
The source for the custom of not cutting nails on Rosh Chodesh is from the testament of Rabbi Yehudah Ha-Chassid, which is cited as halachah by the Be’er Heitev to Orach Chaim 260:2.
However, other halachic authorities do not mention the prohibition, for instance the Shulchan Aruch Ha-Rav (see Badei Ha-Shulchan 73:4 who notes this).
The Aruch Ha-Shulchan (Orach Chayim 260:6) comments that since the custom has no basis in halachah (meaning that it is not found in any Talmudic source; Rabbi Yehudah Ha-Chassid warns against it for reasons of safety based on esoteric considerations), there is no special reason to adopt the custom and one is not more demanding in his religious observance for keeping it.
Therefore, if there is a special reason for cutting nails, such as hygienic considerations, it is certainly permitted to cut them. The same is true for cutting nails on Friday, which is customary for many.
There is no need to do “hataras nedarim” beforehand (Shach, Yoreh De’ah 214:1; Dagul Merevava, Yoreh Deah 214).
See also Or Le-Zion (vol. 3); the Radbaz also notes the custom of taking haircuts on Rosh Chodesh, indicating that the proscription of taking a haircut was not accepted in his community. See also Orchos Chaim (260:3) who writes that on Erev Shabbos one can certainly cut nails even on Rosh Chodesh.