Keil Erech Apayim:
The prayer of Keil Erech Apayim is recited on Mondays and Thursdays prior to taking out the Sefer Torah.
Fast day: Some are accustomed to recite Keil Erech Apayim prior to Kerias Hatorah on fast days, even if it does not fall on Monday or Thursday. Some are accustomed to recite it even by Mincha. Practically however, the widespread custom is to only recite the prayer by Shacharis of Mondays and Thursdays prior to opening the Aron, and not on a fast day that falls on a different weekday. It is likewise not recited by Mincha of a fast day even if it falls on Monday or Thursday.
 Mentioned in: Mateh Moshe 221 [prior to Kerias Hatorah]; Abudarham Seder Shacharis Shel Chol [as part of Monday and Thursday Tachanun]; Kol Bo 13 Midos Utechinos [As continuation of Tachanun]; Peri Eitz Chaim Shaar Kerias Sefer Torah 5 [As continuation of Tachanun]; Maharil “Bein Pesach Leshavuos”; Shaar Hakolel 11/14
The reason: It is said prior to Kerias Hatorah as it asks forgiveness for having transgressed what we are to read in the Torah. [Mateh Moshe ibid] Others write the purpose of this prayer is to serve as a replacement for the long prayer of Vehu Rachum that is recited on Mondays and Thursday’s for those that come late to Shul. It thus has no relevance on days that the long Vehu Rachum is not recited. [Otzer Dinim ibid]
 Custom of Maharil and Rebbe Meir Shatz brought in Maharil “Between Pesach and Shavuos” p. 21; Implication of Arizal in Peri Eitz Chaim ibid that it is recited “before Kerias Hatorah”
 Abudarham, brought in Likkutei Maharich 3/49
 Likkutei Maharich ibid; Otzer Dinim Uminhagim p. 16; Rav Asher Lemel Hakohen; Implication of Abudarham ibid which lists this prayer as part of the Tachanun of Mondays and Thursdays.
The reason: As the purpose of this prayer is to serve as a replacement for the long prayer of Vehu Rachum that is recited on Mondays and Thursday’s for those that come late to Shul. It thus has no relevance on days that the long Vehu Rachum is not recited. [Otzer Dinim ibid]
 Likkutei Maharich ibid
 M”A 131/1; Kneses Hagedola 134/3; Olas Tamis 134/2; M”B 134/4; Kitzur SHU”A 22/10; Kaf Hachaim 134/4