That old tune wormed its way into my head yesterday and I wondered "what the hell is paddywhack?" ("And a knicknack paddywhack, give the dog a bone...") Here's what Wikipedia says about it.
Paddywhack (also spelt Paddywack) or Nuchal ligament/Ligamentum nuchae, is a strong elastic ligament or tendon in the midline of the neck of sheep or cattle which relieves the animal of the weight of its head. It is pale yellow in colour. (The yellow colour is the elastin on the ligaments.) The name is derived from the corruption of paxwax (originally faxwax Old English hair + to grow).Ugh. I'm sorry I looked it up. Sometimes it's better to be ignorant.
It is eaten in several countries. It is high in protein (78%), fat (10%), crude fibre (0.7%), and crude ash (1%). The meat is taken from domestic cattle, the bison, African buffalo, the water buffalo, the yak, and the four-horned and spiral-horned antelopes.
Dried paddywhack is commonly packaged and sold as a dog treat.