Short prayer after a vort or dvar torah

Question

I've recently moved to a new area and the Shul that I now attend will often give a vort at the end of Shacharit and Maariv before allowing mourners to say another Kaddish. I've never heard it before and they say it quite quickly but the words I can make out are "Rabbi Hanina". Apologies for the lack of information but hoping someone would be able to help me find out what they are saying and the significance of it. Thanks in advance!

Answer 1

You are thinking of the famous saying of chazal (our Rabbis of blessed memory) found in Makkos 23b and famous for being mentioned at the end of Pirkei Avos:

רִבִּי חֲנַנְיָא בֶּן עֲקַשְׁיָה אוֹמֵר רָצָה הַקָּדוֹשׁ בָּרוּךְ הוּא לְזַכּוֹת יִשְׂרָאֵל לְפִיכָךְ הִרְבָּה לָהֶם תּוֹרָה וּמִצְוֹת שֶׁנֶּאֱמַר יְיָ חָפֵץ לְמַעַן צִדְקוֹ יַגְדִּיל תּוֹרָה וְיַאְדִּיר

Rebbi Chaninah/Chananyah (there are different versions) ben Akashiah says: The Holy One, praise to Him, wanted to increase the merit of Israel; therefore He multiplied for them teaching and commandments, as it is written, the Eternal desires for the sake of His justice to make His instruction great and prodigious.

After the mishnah the mourners can say a kaddish d'rabbanan.

Rashi notes on it mentioning at the end of pirkei avos the following:

לא אמר למלתה גבי מסכת אבות, אלא במסכת מכות באלו הן הלוקין (כ"ג ע"ב). לפי שיש בה סיום נאה, נהגו כל העם לאמרו בסוף כל פרק ופרק. לפי שאין אומרים קדיש על המשנה, אלא האדגה

This matter is not said exclusively by Tractate Avos, but rather in Tractate Makkos in the chapter entitled 'These are the ones who are lashed' 23b, as it has a nice ending, they became accustomed for every person to say this after each and every chapter. Since one does not say Kaddish over a mishnah but rather over a piece of aggada.

This practise is codified in halacha and is brought down in the Magen Avraham 54:3 and Mishnah Berurah 54:9

Answer 2

From https://outorah.org/p/4589/

The Rabbis’ Kaddish (“Kaddish D’Rabbanan”) is recited after a public lecture in the Oral Torah. There is an opinion that the primary practice is to recite this Kaddish after learning Aggadah, the non-legal portions of the Talmud or Midrash. This is based on Sotah 49a, which tells us that since the Temple was destroyed, the world endures because of the recitation of Kedusha in Uva L’Tziyon and the “yehei shmei rabbah” in the Kaddish after Aggadah. Accordingly, the practice has become to recite the Aggadic passage of “Rabbi Chananya…” (Makkos 23b) between a public Torah lecture and Kaddish. Since it is related to Torah study, the Rabbis’ Kaddish includes a passage on behalf of “the Jews, the teachers, their students, their students’ students, and all those who engage in the study of Torah.”