In a case where there is a minyan on a fast day, but one of those required in order to be able to read from the Torah accidentally ate (i.e. there are 5 fasters, 4 non-fasters, and this one who ate but accidentally). Are they allowed to read from the Torah?
ב"ה Shalom The Mishna B'rura says that if the fast falls on Monday or Thursday which in any case there is reading from the Torah , IN THE MORNING it is enough that there are 6 people fasting. However, for the mincha reading and in the mornings of other days besides Monday or Thursday, there should be ten people fasting. In the event that there are only 7 fasting and the others are not fasting because of justified health reasons (not just because they didn't want to) the Chazan can say anennu and the Torah can be read. (משנה ברורה סימן תקסו ס”ק יד). Some interpret the Mishna B'rura that even if there were 6 people fasting. ( פסקי תשובות תקס"ו: ד, הערה 17) There is also an opinion of some very important Rabbis, that even if there are only 3 people fasting, and the others are not fasting because of justified health reasons, as we said before, the Chazan can say anenu and the Torah can be read. (ערוך השולחן תקס"ו:ז, שו"ת צמח צדק אורח חיים סי' קי, שו"ת ציץ אליעזר חי"ד סי' נ"ו( However, this is not the practiced custom. So therefore, one should not plan a fast day minyan in advance upon such a situation when only 3 are fasting, but if it just happens to work out that way and there is no other available minyan, there is who to rely upon. So in the case you mentioned, according to the Mishna Brura you cannot read the torah at mincha, but in the extreme case that it just happened to be that way there are those who allow the torah reading. All the best