What is the name of the flute that has a mouthpiece projecting from its body and consists of multiple finger holes?

The flute described in the question is best represented by the ocarina. The ocarina is a type of wind instrument that typically features a mouthpiece protruding from its body and is designed with several finger holes that allow the player to produce different notes by covering or uncovering these holes. It is commonly made from ceramic or plastic and has a distinct, rounded shape.

In contrast, the tarkas is a traditional South American flute that usually does not have a projecting mouthpiece, and it often has a more cylindrical shape. The rasp, while a musical instrument, is not classified as a flute and typically produces sound through scraping rather than air. The siku, often referred to as panpipes, consists of multiple tubes of varying lengths and is played by blowing across the open ends, which makes it different from a single body flute like the ocarina.

Thus, the ocarina's unique construction aligns perfectly with the description provided in the question.