1. Question: [Thursday, 4th Tishreiy, 5782]
By Mincha time I remembered that by Shacharis I forgot to recite Hamelech Hakadosh. What should I do?
After Mincha you should Daven an extra Shemoneh Esrei as a Tashlumin for Shacharis.
Sources: See Admur 108:17; M”A 108:16; Elya Raba 116:16; Yad Aaron; Machazik Bracha 108:9; Kesher Gudal 20; Mamar Mordechai 108:12; Ashel Avraham 108:16; Shalmei Tzibur p. 172; Beis Menucha 4; Kaf Hachaim 108:54; Or Letziyon 4:18
2. Question: [Thursday, 4th Tishreiy, 5782]
When Davening Shemoneh Esrei a second time after Mincha as Tashlumin for Shacharis, should one recite Ashreiy inbetween?
Yes, one should recite Ashreiy, however, stipulating that he is not reciting Ashreiy as an obligation but rather as a voluntary reading of verses from scripture.
Explanation: The general law is that whenever one prays two Shemoneh Esrei is back to back, they should be separated with the recital of Ashreiy, in order to pray the second Shemoneh Esrei after words of consolement of Torah. Nonetheless, according to the Zohar, it is not proper to ever recite the Psalm of Ashreiy/Tehilas Ledavid after Mincha or Maariv as an obligation prayer, and rather it is only to be read like one who reads from Scripture. Accordingly, we concluded as above that it is to be said not in a form of obligation
Sources: See Admur 108:5; 623:1; Michaber 234:2; 623:1; Rama 108:2; Zohar 3:226; Taz 108:3; M”A 108:5; Bach 108; Elya Raba 108:2-3; Levush 108; Machazik Bracha 108:5; Kesher Gudal 22:12; Rivash 140; Hagahos Semak 11:44; Chesed Lealafim 108:1; Ben Ish Chaiy Mishpatim 9; Shaar Hakolel 44:19; Kaf Hachaim 108:20; Likkutei Sichos 18:351 footnote 7