NEWARK, N.J.—Can this jar of gefilte fish be saved?

A staple of Jewish cuisine, gefilte fish—Yiddish for stuffed fish—has been a fixture for Manischewitz Co., which started selling jars of the beige dumplings of minced carp, pike and whitefish in the late 1940s. Families still snap up the jars for holiday dinners, such as Rosh Hashana, the Jewish New Year, which begins Wednesday evening.

Traditional...