Does there exist an English word of Yiddish origin meaning “bothersome busywork” (admittedly redundant)?
Is that what “chazzerai” means? (I’m sure that's spelled incorrectly)
Leo Rosten, Hooray for Yiddish: A Book About English (1982) offers the following entry for khazer:
Khazer (standard) | khazeray | chazzer | chozzer
[Pronunciation:] Rattle the kh: KHAH-z'r and kha-z'RY. Hebrew: pig.
[Meanings:] 1. Pig. 2. A greedy lout; a glutton. 3. An ungrateful type. 4. A cheap selfish person. 5. One who takes advantage of another, through cheap tricks or cunning tactics.
Oddly enough, given the ancient taboo on the porcine, Jews rarely use khazer to describe someone dirty, as English speakers do when they snap "Pig!"
Kahazerish, the adjective, is used (also oddly enough) with a tinge of envy: "He lives a khazerish tug" means "He's living it up."
Khazeray means awful food, cheap merchandise, obscenity, or contemptible reading. I much prefer khazeray to its colloquial synonym, "crap." Its sonic lilt is enough to recommend it.
So if the word you heard as "chazzerai" is the same one that Rosten identifies as khazeray, it has far broader application in Yiddish than is implied by the specific sense "bothersome busywork." Certainly one might refer to "bothersome busywork" as "crap" (I think that I have, on more than one occasion)—but the world is full of things that qualify colloquially as "crap."
With regard to whether khazeray has crossed over from Yiddish to English to any meaningful extent, I think that the answer is unequivocally no. For evidence that this is the case, I point to the fact that Rosten, in his book The Joys of Yinglish 1989)—which advertises itself as "An exuberant dictionary of Yiddish words, phrases and locutions that are now part of the English language or, because of their incomparable connotations, should be"—contains no entry for khazer or khazeray (in any spelling). Evidently, the fact that khazeray is comparable to "crap" goes a long way toward rendering it dispensable in English.