A Minyan with less than a minyan fasting - Yechaveh Daat 1:89

Rav Yosef deals here with the question of whether or not a minyan should layn on a fast day if only six members of the minyan are fasting. He begins by quoting Rashba and the Shulchan Aruch who say that in such a situation the Shliach Tzibbur does not say Aneinu and that this applies to layning as well. However, he then cites the Maharam ben Chaviv who differentiates between the fast days that are fixed into the calendar, when having six people fasting would be sufficient, and fast days that are declared in response to a specific calamity, when a full minyan of fasting individuals would be needed. This psak seems to have been fairly widely accepted among Sephardic poskim and in Israel.

As an add-on, Rav Yosef deals with the issue of making Birchot haTorah for non-standard layning times. For example, if a shul reads the Purim morning Torah reading on both the 14th and 15th of Adar due to some doubt, they may make the brachot on both days. Rav Yosef cites the custom of reading the Torah with brachot on Simchat Torah night as proof - even though this reading is not halachically necessary, since it is an accepted minhag the brachot can be made. Rav Yosef does note that both the Chida and the Netziv oppose this view, but he himself favors it.