The following prayer יְהִי רָצוֹן , composed by Rav, head of the Babylonian yeshiva at Sura, is mentioned in the Talmud, ( Maseches Berachos 16b), but not as a prayer specifically ordained for blessing the new month. It was actually a daily prayer that Rav recited each day after the regular service.
The יְהִי רָצוֹן was adapted by Ashkenazi Jews as a prayer for the coming of the new month by adding the words: שֶׁתְּחַדֵשׁ עָלֵינוּ אֶת הַחוֹדֶשׁ הַזֶּה לְטוֹבָה וְלִבְרָכָה but Sephardic Jews omit the entire prayer as it is not mentioned either by Rambam or Avudraham.
יְהִי רָצוֹן מִלְּ֒פָנֶֽיךָ May it be Your will,
יְהֹוָה אֱלֹהֵֽינוּ וֵאלֹהֵי אֲבוֹתֵֽינוּ Adonoy, our God and God of our fathers,
שֶׁתְּ֒חַדֵּשׁ עָלֵֽינוּ אֶת הַחֹֽדֶשׁ הַזֶּה to renew this [coming] month for us
לְטוֹבָה וְלִבְרָכָה. for goodness and blessing.
וְתִתֶּן לָנוּ חַיִּים אֲרוּכִים Grant us long life—
חַיִּים שֶׁל שָׁלוֹם a life of peace,
חַיִּים שֶׁל טוֹבָה a life of goodness,
חַיִּים שֶׁל בְּרָכָה a life of blessing,
חַיִּים שֶׁל פַּרְנָסָה a life of sustenance,
חַיִּים שֶׁל חִלּוּץ עֲצָמוֹת a life of bodily vigor,
חַיִּים שֶׁיֵּשׁ בָּהֶם a life in which there is
יִרְאַת שָׁמַֽיִם וְיִרְאַת חֵטְא fear of Heaven and fear of sin
חַיִּים שֶׁאֵין בָּהֶם a life in which there is no
בּוּשָׁה וּכְלִמָּה shame or disgrace,
חַיִּים שֶׁל עֹֽשֶׁר וְכָבוֹד a life of prosperity and honor,
חַיִּים שֶׁתְּ֒הֵא בָֽנוּ a life in which there will be
אַהֲבַת תּוֹרָה וְיִרְאַת שָׁמַֽיִם love of Torah and fear of Heaven,
חַיִּים a life [in which]
שֶׁיִּמָּלְ֒אוּ מִשְׁאֲלוֹת לִבֵּֽנוּ will be fulfilled the wishes of our hearts
לְטוֹבָה. אָמֵן סֶֽלָה: for goodness. Amein, Selah.
The following prayer, is the actual blessing of the month. The Chazzan takes the Torah in his hands and declares:
מִי שֶׁעָשָׂה נִסִּים לַאֲבוֹתֵֽינוּ He Who performed miracles for our fathers,
וְגָאַל אוֹתָם מֵעַבְדוּת לְחֵרוּת. and redeemed them from slavery to freedom,
הוּא יִגְאַל אוֹתָֽנוּ בְּקָרוֹב may He redeem us soon
וִיקַבֵּץ נִדָּחֵֽינוּ and gather our dispersed
מֵאַרְבַּע כַּנְפוֹת הָאָֽרֶץ. from the four corners of the earth;
חֲבֵרִים כָּל יִשְׂרָאֵל. united in friendship is all of Israel!
וְנֹאמַר אָמֵן: And let us say, Amein.
רֹאשׁ חֹֽדֶשׁ (פלוני) The new month of (name the month),
יִהְיֶה בְּיוֹם (פלוני) which will be on (name the day or days)
הַבָּא עָלֵֽינוּ וְעַל כָּל יִשְׂרָאֵל will come to us and all Israel
לְטוֹבָה: for goodness.
יְחַדְּ֒שֵֽׁהוּ May He renew it,
הַקָּדוֹשׁ בָּרוּךְ הוּא. the Holy One, blessed is He,
עָלֵֽינוּ וְעַל כָּל עַמּוֹ for us and for all His people,
בֵּית יִשְׂרָאֵל. the House of Israel,
לְטוֹבָה וְלִבְרָכָה. for goodness and blessing
לְשָׂשׂוֹן וּלְשִׂמְחָה. for happiness and for rejoicing,
לִישׁוּעָה וּלְנֶחָמָה. for deliverance and for consolation,
לְפַרְנָסָה טוֹבָה for plentiful sustenance,
וּלְכַלְכָּלָה. and support,
לְחַיִּים טוֹבִים וּלְשָׁלוֹם. for good life and for peace,
לִשְׁמוּעוֹת טוֹבוֹת. for good tidings,
וְלִבְשׂוֹרוֹת טוֹבוֹת. and for good news,
וְלִגְשָׁמִים בְּעִתָּם. for rain at the proper time,
וְלִרְפוּאָה שְׁלֵמָה. for total recovery,
וְלִגְאֻלָּה קְרוֹבָה. and for early redemption,
וְנֹאמַר אָמֵן: and let us say, Amein.