The prayer Avinu Malkeinu, is first mentioned in the Talmud, making it one of the most ancient parts of the HHD liturgy. Rabbi Akivah only offers two prayers, though later versions expand it to 5, 22 and as many 53. It's said after the Amidah (the main prayer, said while standing) and before the Torah service. It is recited throughout the Ten Days of Awe, from Rosh Hashanah through Yom Kippur, as well as on fast days.
It says: Our Father, Our King be gracious and answer us, though we have nothing to show for ourselves. Do deeds of tzeddakah (generosity) and chesed (loving kindness) through us. And save us.
שׁוּב מַעֲשֶׂה בְּרַבִּי אֱלִיעֶזֶר שֶׁיָּרַד לִפְנֵי הַתֵּיבָה וְאָמַר עֶשְׂרִים וְאַרְבַּע בְּרָכוֹת וְלֹא נַעֲנָה יָרַד רַבִּי עֲקִיבָא אַחֲרָיו וְאָמַר אָבִינוּ מַלְכֵּנוּ אֵין לָנוּ מֶלֶךְ אֶלָּא אָתָּה אָבִינוּ מַלְכֵּנוּ לְמַעַנְךָ רַחֵם עָלֵינוּ וְיָרְדוּ גְּשָׁמִים הֲווֹ מְרַנְּנִי רַבָּנַן יָצְתָה בַּת קוֹל וְאָמְרָה לֹא מִפְּנֵי שֶׁזֶּה גָּדוֹל מִזֶּה אֶלָּא שֶׁזֶּה מַעֲבִיר עַל מִידּוֹתָיו וְזֶה אֵינוֹ מַעֲבִיר עַל מִדּוֹתָיו
There was another incident involving Rabbi Eliezer, who descended to serve as prayer leader before the ark on a fast day. And he recited twenty-four blessings, but he was not answered. Rabbi Akiva descended before the ark after him and said: Our Father, our King, we have no king other than You. Our Father, our King, for Your sake, have mercy on us. And rain immediately fell. The Sages were whispering among themselves that Rabbi Akiva was answered while his teacher, Rabbi Eliezer, was not. A Divine Voice emerged and said: It is not because this Sage, Rabbi Akiva, is greater than that one, Rabbi Eliezer, but that this one is forgiving, and that one is not forgiving. God responded to Rabbi Akiva’s forgiving nature in kind by sending rain.
two translations- the first from the commonly used in Ashkenazi communities, Machzor Rosh Hashanah Ashkenaz, the second from Makhzor Ktiray Or by R. David Wolfe-Blanke
אָבִֽינוּ מַלְכֵּֽנוּ חָטָֽאנוּ לְפָנֶֽיךָ : Our Father, our King! we have sinned before You. 1
אָבִֽינוּ מַלְכֵּֽנוּ אֵין לָֽנוּ מֶֽלֶךְ אֶלָּא אָֽתָּה: Our Father our King! we have no King except You. 2
אָבִֽינוּ מַלְכֵּֽנוּ עֲשֵׂה עִמָּֽנוּ לְמַֽעַן שְׁמֶֽךָ: Our Father, our King! deal with us [kindly] for the sake of Your Name. 3
אָבִֽינוּ מַלְכֵּֽנוּ חַדֵּשׁ עָלֵֽינוּ שָׁנָה טוֹבָה: Our Father, our King! renew for us a good year 4
