Bris of a Mamzeir:[1]

A Mamzer child, which is defined as a child who was born from an Erva defined relation [i.e. of adultery or incest], is considered like a regular Jew and is hence to be circumcised as usual.

The blessings and ceremony:[2] All of the usual blessings are to be recited upon doing the circumcision, as is usually done, until the words Koreis Habris [i.e. Al Hamila, Lehachniso Beveriso Shel Avraham Avinu, Hagafen, Asher Kidesh Yedid Mibeten]. However, the prayer of Kayeim Es Hayeled Hazeh which is normally recited after the blessing of Koreis Habris, is not to be recited as one is not to supplicate on behalf of such a child.[3] [Likewise, those in attendance of the circumcision are not to recite the blessing of Kisheim Shenichnas.[4]]

Publicizing that the child is a Mamzer:[5] One is to publicize at the time of the circumcision that the child is a Mamzer.

Naming the child: Some Poskim[6] write that a Mamzer is to be named the name “Kidur” which is a play on the verse which describes a flipped generation.

Where to perform the circumcision: Some Poskim[7] write that the circumcision of a Mamzer should not take place in the regular location where circumcisions take place, such as rather than circumcising the child inside of the Shul he should rather be circumcised by the entrance of the Shul.

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[1] Michaber 263:4; Pesakim Uteshuvos 265:14

[2] See Encyclopedia Talmudit Vol. 4 Erech Bris Mila p. 257 footnotes 177-179

[3] The reason: As the Jewish people who are holy do not desire for the Mamzer population to become established. [Shach 263:9]

[4] Shach 263:9 in name of Maharil

[5] Rama 263:4

[6] Taz 263:8 in name of Maharil

[7] Taz 263:8 and Shach 263:11 in name of Maharil