In Shacharit and Minchah of public fast days, we read the Torah section that describes how God forgave the Jews for the sin of the golden calf (Tractate Sofrim 17:7, SA 566:1). This symbolizes that just as God forgave us for the sin of the calf and gave us a new set of tablets, so will He forgive all of our sins and rebuild the HolyTemple, speedily in our days.
Most poskim hold that we read the haftarah of “Dirshu HaShem” (Yeshayah 55) at Minchah, and all Ashkenazim follow this practice (Rama 566:1). However, most Sefardim do not read a haftarah. Nonetheless, a Sefardic Jew who is called up third to the Torah in a place where they read the haftarah should read it with its blessings (Yaskil Avdi 6:9; see Torat HaMo’adim 4:2).
These readings are read only in a place where at least six people are fasting… One should only call a person who is fasting up to the Torah on a fast day. If, however, someone who is not fasting was [accidentally] called up to the Torah, and he is embarrassed to say that he failed to fast, he may go up to the Torah.
Ashkenazim recite the Avinu Malkeinu prayer after the Shemoneh Esrei of both Shacharit and Minchah, while Sefardim do not.