(א)
V1.
סְלַח לָֽנוּ אָבִֽינוּ 1 כִּי חָטָֽאנוּ 2
V2.
סְלַח לָֽנוּ אָבִֽינוּ כִּי חָטָֽאנוּ 3
V1.
מְחַל לָֽנוּ מַלְכֵּֽנוּ 4 כִּי פָשָֽׁעְנוּ 5
מְחַל לָֽנוּ מַלְכֵּֽנוּ כִּי פָשָֽׁעְנוּ 6
כִּי מוֹחֵל וְסוֹלֵֽחַ אָֽתָּה:
בָּרוּךְ אַתָּה יְהֹוָה חַנּוּן הַמַּרְבֶּה לִסְלֽוֹחַ:
(1) Pardon us, our Father, for we have sinned, forgive us, our King, for we have transgressed; for You forgive and pardon. Blessed are You, Adonoy, Gracious One, Who pardons abundantly.
In this beracha, we ask Hashem for forgiveness. Our relationship with Hashem is a blend of Him being our compassionate Father in Heaven and simultaneously as our Ruler and King.
We say this sentiment in Mussaf to Rosh Hashana:
הַיּוֹם הֲרַת עוֹלָם הַיּוֹם יַעֲמִיד בַּמִּשְׁפָּט כָּל יְצוּרֵי עוֹלָמִים. אִם כְּבָנִים אִם כַּעֲבָדִים. אִם כְּבָנִים רַחֲמֵנוּ כְּרַחֵם אָב עַל בָּנִים. וְאִם כַּעֲבָדִים עֵינֵינוּ לְךָ תְּלוּיוֹת עַד שֶׁתְּחוֹנֵנוּ וְתוֹצִיא כָאוֹר מִשְׁפָּטֵנוּ, אָיֹם קָדוֹשׁ:
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The Tur, by connecting the dots between two different pesukim, explains why.
The first passuk, discussing Yeshayahu's urging Klal Yisrael to teshuva, says:
(ז) יַעֲזֹ֤ב רָשָׁע֙ דַּרְכּ֔וֹ וְאִ֥ישׁ אָ֖וֶן מַחְשְׁבֹתָ֑יו וְיָשֹׁ֤ב אֶל־יְהֹוָה֙ וִֽירַחֲמֵ֔הוּ וְאֶל־אֱלֹהֵ֖ינוּ כִּֽי־יַרְבֶּ֥ה לִסְלֽוֹחַ׃
The sinful, their plans;
Let each one turn back to GOD
So as to be pardoned;
To our God,
Who freely forgives.
(יג) כְּרַחֵ֣ם אָ֭ב עַל־בָּנִ֑ים רִחַ֥ם יְ֝הֹוָ֗ה עַל־יְרֵאָֽיו׃
so the LORD has compassion for those who fear Him.
The second passuk pairs "mercy" with a "father."
Using the transitive property, "forgiveness" is a term that fits in the context of a "father."
What emerges from here, is that Hashem's mercy towards us in forgiving our sins, is likened to a loving father forgiving his child's mistakes.*
*This comparison offers comfort as it reflects a close, nurturing mercy that promotes our well-being as His children, rather than mere pity shown to a needy individual, which lacks the same depth of love.
ומה שתיקנו לומר אבינו בברכת השיבנו ובסלח לנו מה שאין כן בשאר הברכות...ובסלח לנו משום הא דכתיב וישוב אל ה' וירחמהו ואל אלהינו כי ירבה לסלוח ע"כ אנו מזכירין רחמי האב כדכתיב כרחם אב על בנים שירחם עלינו ויסלח לנו.
The reason is because the word "chet" (the root of chatanu) is used in the same context as the term we use for "forgiveness"(selach) in a passuk in Melachim.
The passuk is was said by Shlomo HaMelech when he dedicated the Beis HaMikdash. After thanking Hashem for fulfilling the desire of his father Dovid HaMelech to build the Beis HaMikdash, he goes on to make many pleas from Hashem. One of his pleas is that He forgive Klal Yisrael's sins when they do teshuva before Him.
(נ) וְסָלַחְתָּ֤ לְעַמְּךָ֙ אֲשֶׁ֣ר חָטְאוּ־לָ֔ךְ וּלְכׇל־פִּשְׁעֵיהֶ֖ם אֲשֶׁ֣ר פָּשְׁעוּ־בָ֑ךְ וּנְתַתָּ֧ם לְרַחֲמִ֛ים לִפְנֵ֥י שֹׁבֵיהֶ֖ם וְרִחֲמֽוּם׃
סְלַח לָֽנוּ אָבִֽינוּ כִּי חָטָֽאנוּ 3
Avudraham brings an alternate inflection to read, "Forgive us Our Father even though we have sinned."
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מְחַל לָֽנוּ מַלְכֵּֽנוּ 4 כִּי פָשָֽׁעְנוּ 5
Avudraham offers an explanation for this clause, (which also sheds light on the terminologies used in the first clause.)
See Introduction to this beracha for the basic concept behind the reason as to why there is a need switch the terminologies between asking for forgiveness from a king as opposed to a father.
1. "Chet" denotes a minor infraction, whereas "Pesha" denotes a more grievous offense.
2. "Selicha" denotes a gentle request to pardon a sin, whereas "Mechila" denotes a valiant effort at a sincere attempt at reconciliation from an exacting person to accept ones apology.
In the first clause, we are asking forgiveness from our Father, and just like ones father is (usually) forgiving if his son would affronts him, and considers the degree of a deliberate offense as though it was unintentional (which translates into the category of "chet"), therefore the intensity of the forgiveness needed to ask from him is also lower (translating to the category of "selicha").
However, in the second clause we are asking forgiveness from our King, and with a king the opposite is true. Even an unintentional offense is treated with the gravity of an intentional offense (speeding tickets are handed out under wither condition!), therefore we ask beseech Hashem forgive (translating to the category of "mechila) our serious transgressions (peshanu).*
In the beginning of the Avudraham's comments, he says that the term pesha is based on the passuk in Melachim, which we brought above as the source for the term chet and its counterpart "selicha."
There, the term "pesha" is used in the context with "selicha" with seemingly no reason to use mechila? Why then does our beracha feel the need to use the term "mechila" if the passuk itself (which Avudraham himself brought) to be used with "pesha"?
(יט) סלח לנו אבינו כי חטאנו על שם וסלחת לעמך אשר חטאו לך. מחול לנו מלכנו כי פשענו על שם ולכל פשעיהם אשר פשעו בך. וסמך סליחה וחטא לאב ומחילה ופשע למלך מפני שכל זדונות שעושה הבן דומות לפני האב כשגגות ונקלים יהיו בעיניו לסלחם כמו שחטא נקל מפשע, ולפני המלך דומות שגגות העם כזדונות כמו שנאמר מלך במשפט יעמיד ארץ ולכן יש לומ' לו כי פשענו לפי שפשע גדול מחטא, ומחילה יש לבקש מכל אדם המקפיד ומדקדק על חבירו או על עבדו שימחול לו עלבונו אבל אין לומר שיסלח לו עלבונו, ולכן יש לומ' לאב שאינו מקפיד ומדקדק כל כך סלח לנו כלומ' שאף הקפדה מועטת לא תשאר.
מְחַל לָֽנוּ מַלְכֵּֽנוּ כִּי פָשָֽׁעְנוּ 6
Avudraham brings an alternate inflection to read, "Forgive us Our King even though we have sinned."