The following is from Daily Halacha by Rabbi Eli Mansour:
Rabbi David Halevi, 1658-1667) comments that when a Brit Milah (ritual circumcision) is to be performed on Rosh Hashanah, it is held just prior to the sounding of the Shofar. Both Brit Milah and Shofar have the capacity to arouse Divine compassion upon the Jewish people, and it is therefore proper to combine them in this manner in our effort to invoke God’s mercy on this day of judgment.
The Taz then proceeds to record a custom he observed that further underscores the importance of this association between Brit Milah and Shofar. The final stage of Brit Milah is “Mesisa Be’fe,” when the Mohel (person who performs the circumcision) draws some blood from the infant’s wound with his mouth, in order to prevent infection.* The Taz observed in situations of a Brit performed on Rosh Hashanah that one of his Rabbis, Rabbi Feivel, would have the Mohel blow the Shofar after the Brit without first cleaning his mouth. Rabbi Feivel wanted the blood from the Brit to come in direct contact with the Shofar, in order to reinforce this association between the two Mitzvot of Milah and Shofar.
It should be noted, however, that this practice may be followed only if the Mohel will himself be sounding the Shofar. In most cases, however, when the person who performs the circumcision is not the same one who sounds the Shofar, the one sounding the Shofar may not perform the Mesisa in order to fulfill this custom. As the Hatam Sofer (Rabbi Moshe Sofer of Pressburg, 1762-1839) and Hochmat Adam (Rabbi Avraham Danzig of Vilna, 1748-1820) write, drawing blood is generally forbidden on Shabbat and Yom Tov, and it is allowed only in the context of a Brit Milah, which overrides the prohibitions of Shabbat and Yom Tov. Hence, only the Mohel, who performed the actual circumcision, may perform the Mesisa on Yom Tov. Therefore, the person sounding the Shofar may not perform the Mesisa unless he had performed the circumcision. Although the practice recorded by the Taz is certainly an admirable one, it is followed only if the same person who performs the circumcision will also sound the Shofar.
Summary: If a Brit Milah is performed on Rosh Hashanah, it is held immediately preceding the Shofar blowing. If the person who performs the circumcision will also be sounding the Shofar, then he should not clean his mouth after performing the Mesisa (drawing blood from the infant’s wound), so that the blood of the circumcision will touch the Shofar.
The same teaching is also found in Why Aren't Jewish Women Circumcised, which goes on to say:
"In contrast, R. David Abudarham (Fourteenth Century)... says that a circumciser must wash his hands and mouth before reciting the benedictions, so that he may utter the prayers, 'in cleanliness'."
* Note: Also transliterated as metzitzah b'peh or mezizah, some rabbinical authorities have ruled that this traditional practice of direct contact should be replaced by using a glass tube between the wound and the mohel's mouth to conform to contemporary sanitation practices.
For more on circumcision and shofar...
Illustration from Sefer Minhagim (Book of Customs), Amsterdam, 1722