In general, when Rosh Hodesh, Hanukkah, a public fast, etc. fall on a weekday, the Sephardic custom is to recite the Mizmor designated for that day in addition to Bet Ya’akov, Shir Hama’alot, Shir Shel Yom and Hoshienu. There is a lesser known custom among Sepharadim, including Moroccans, however, to omit Bet Ya’akov, Shir Hama’alot, Shir Shel Yom and Hoshienu and to only recite the special Mizmor.
One may ask why the Shir Shel Yom as well as the Mizmor for the special day are not both recited. The HIDA (Birkei Yosef 693:3) quotes the Yerushalmi (Shekalim and Sukkah), which says that once one Mizmor is recited it abrogates the other one. Rabbi Yosef Hazan (Shu”t Hikre Lev, vol. 2, Orah Haim, § 32) explains at length that in the times of the Bet HaMikdash, the Levi’im would sing the Shir Shel Yom, unless there was another occasion such as Rosh Hodesh, in which case the Mizmor for that day would be sung. Furthermore, he says that the nowadays, we do not recite the Shir Shel Yom to emulate exactly what was done in the Bet HaMikdash, but rather to highlight the specialness of each day of the week. Therefore, if another special day takes place, the Mizmor specific to that day is recited and there is no reason to add on the Shir Shel Yom. As such, Rabbi Yedidya Monsonego (Pirhe Shoshanim, Hilchot Tefila § 14), Rabbi David Ovadia (Nahagu Ha’am, Hilchot Tefila § 25), Rabbi Raphael Ben Shimon (Nahar Mitzrayim, Hilchot Tefila § 11) and Rabbi David HaKohen-Skali (Kiryat Hana David, vol. 2, § 20) write that the custom in Morocco and Egypt was to omit the Shir Shel Yom when the day coincided with a Rosh Hodesh, Hanukkah, a public fast, Purim or if a groom was present, and to recite the Mizmor specific to that day only. There were communities in Morocco that included the Shir Shel Yom, Bet Ya’akov, etc., but they were in the minority.
Summary: The Shir Shel Yom, along with Bet Ya’akov, Shir Hama’alot and Hoshienu are omitted on Rosh Hodesh, Hanukkah, Purim, a Ta’anit or when a groom is present, and only the Mizmor specific for that day is recited.