This question arises whenever instructions state to omit the Penitential Rite. Oftentimes, priests, answering this question in the affirmative, will also omit the Kyrie. However, there is evidence to suggest that, in fact, the Kyrie is not a part of the penitential rite. Thus, priests omit the Kyrie when it should be included. Here are several citations that indicate that the Kyrie is separate from the penitential rite:

GIRM #46.

The rites that precede the Liturgy of the Word, namely, the Entrance, the Greeting, the Penitential Act, the Kyrie, the Gloria in excelsis (Glory to God in the highest) and Collect, have the character of a beginning, an introduction, and a preparation.

GIRM #52.

After the Penitential Act, the Kyrie, eleison (Lord, have mercy), is always begun, unless it has already been part of the Penitential Act.

Ceremonial of Bishops #255, on Ash Wednesday:

The introductory rites of the Mass and, as circumstances suggest, also the Kyrie are omitted, and the bishop immediately says the opening prayer.

GIRM #125.

The Penitential Act follows. After this, the Kyrie is sung or said, in accordance with the rubrics (cf. no. 52).

Roman Missal, Third Edition, which says for Ash Wednesday (Stational) and Palm Sunday:

Omitting the Introductory Rites and, if appropriate, the Kyrie, he says the Collect of the Mass, and then continues the Mass in the usual way.

The Third Edition of the Roman Missal clarified that the Gloria should be sung at weddings. This creates another situation giving us clear instructions to omit the penitential rite and to sing the Gloria, but making no mention of the Kyrie. I had hoped that this issue would be clarified with the new edition of The Order of Celebrating Matrimony. However, the recently published Order of Celebrating Matrimony states:

The Penitential Act is omitted. The Gloria in excelsis is said according to the rubric of the Roman Missal…

An insert like this: “The Penitential Act and the Kyrie are omitted.” OR “The Kyrie and Gloria in excelsis are said…” would have clarified the issue.

The Bishop’s Committee on the Liturgy developed this analysis of the issue: bislitkyrie

The Extraordinary Form: what does it teach us?

As a reference, at a wedding in the Extraordinary Form (the Mass before Vatican II), the Kyrie follows the entrance chant .  In the extraordinary form, the rite of marriage takes place before the Mass is celebrated.  After the couple give their consent, exchange vows, and the rings are blessed and exchanged, the entrance chant begins. In this example, the Kyrie begins at 8:48.

The community at Musicasacra.com forum discuss this topic here.

In cases of ambiguity such as this, discuss the issue with the pastor and proceed according to his decision.