Meaning of “panegoric”

Paregoric, handwritten, is easily misread as panegoric because r in some hands looks like n. You can see this, for example, in this online transcription of medical records,^{[DOC format]} in which paregoric appears ten times and panegoric three times.

This misspelling has been imitated. It has made its way into peoples’ minds, as evidenced by various publications. Not just the popular or banal:

He finishes the new song “New York” to the pealing cheers of the attendant 200, and crouches at stage-lip, sparking a giant panegoric joss stick. [emphasis added]
 —CMJ New Music Monthly.

But also the literary and scholarly:

An old Danish jester named Yorick
Drank a gallon of pure panegoric;
 “My jokes have been dull,”
 He said, “but my skull
Will one of these days be historic.” [emphasis added]
 —Ogden Nash, EOP p139. Misquoted in “Limericks Y”^{[Wayback cached version]}on AngelFire.com. The Nash original has paregoric.

In 1859 each sold Leach Carrigan one or two bales of cotton and used most of the proceeds to pay off bills for the usual variety of store-bought goods they had purchased during the year: … occasional pharmaceuticals such as “painekillers,” asepheteda, panegoric, or tonic …. [emphasis added]
 —McCurry, Stephanie. Masters of Small Worlds: Yeoman Households, Gender Relations, and the Political Culture of the Antebellum South Carolina Low Country.

Even camphor was believed to be a heart stimulant (Wortley Axe 1902) and when combined with other agents as in panegoric or tinctures, can have profound effects on the respiratory tract. [emphasis added]
 —Booth and McDonald 1986. Equine Veterinary Education, Volume 2, Issue 2.^[PDF]

I wish to take with me a little of thy good Magnesia Panegoric & tincture of rubarb. [emphasis added]
 —Letter by Dolley Payne Todd Madison. Transcribed and proofed by “PJM”, Amy Minton, and Holly Shulman.

All to which not a lot snapped The Nolan of the Calabashes at his whilom eweheart photognomist who by this sum taken was as much incensed by Saint Bruno as that what he had consummed was his own panegoric, and wot a lout about it if it was only a pippappoff pigeon shoot that gracesold getrunner, the man of centuries, was bowled out by judge, jury and umpire at batman’s biff like a witchbefooled legate. Dupe. [emphasis added]
 —Joyce, James. Finnegans Wake.

It is very likely that this misspelling has found its way into your module.

Incidentally, paregoric does not mean “medicine that allays pain”, at least not in a general sense. It means a particular mixture of opium, anise oil, benzoic acid, camphor, glycerin, alcohol, and water, originally compounded to soothe the cough of an asthmatic. Paregoric has other uses, notably the control of diarrhea. (See: “paregoric” entry at Wikipedia.)

Also, paregoric should not be confused with panegyric, which is a sort of eulogy. It is even possible that the resemblance between the two words might have contributed to some misspellings of paregoric as panegoric.

The correct term is paregoric. It's of Greek origin and its meaning is the one given in your question. There is a definition at Etymonline.