Tatābuʿ is a verbal noun, and kaffayhi forms its genitive complement. Kaffay is the genitive dual of kaff, “palm/hand,” with a pronoun referring to the child: “his two hands.”
Bi-khayṭin is a prepositional phrase indicating the cord used to drive the toy. Muwaṣṣal is an adjective, meaning connected or attached. The poet describes a concrete mechanism rather than an unexplained blur of activity.
The sequence is the child's hands → motion through a string → the toy's continuous spinning. Following this cause-and-effect chain shows precisely why the image suits the horse's sustained pace.