What is the meaning of your statement: "a prayer or act of worship intended for its own sake"?

An act of worship intended for its own sake is one that the Sacred Law specifically wills to be performed and cannot be subsumed under another act. This is in contrast to acts not intended for their own sake, such as the tahiyyat al-masjid (greeting of the mosque prayer), which can be subsumed under an obligatory prayer. Scholars have differed on the issue of combining intentions. Some hold that a single act of worship cannot serve for two acts except in certain cases, such as one who intends by their prayer both the obligatory prayer and the tahiyyat al-masjid, or one who intends by their ghusl (ritual bath) to remove both minor and major ritual impurity, or the ghusl for Friday and for major ritual impurity, or one who intends by their tayammum (dry ablution) to remove both ritual impurities. This is permissible because the intent of the Lawgiver is achieved by the occurrence of the action. Thus, the obligatory prayer fulfills the tahiyyat al-masjid, and the removal of major impurity includes minor impurity, and the ghusl for major impurity and for Friday have the same action, as does tayammum.