In Czech & Slovak, "polák" (right-leaning accent on "a" making it a long "aah" sound) means a "Pole" - i.e. a Polish man. The Hungarian "Lengyel" has the same meaning.
Don't set too much store by spelling. After Austrian conquest of 1618, the Czech language was banned, & "Pollak" became the more usual spelling within the Austrian Empire - the "o" sound is short, & German spells this phonetically by doubling the following consonant.
Also, many immigrants to the US & other countries re-spelled their names on arrival. "Polack" & "Pollack" are common variants; in Switzerland you even get "Bollag"!
My family (& maybe your great-great-grandmother's) is probably descended from the 15th-century Rabbi Ya'akov Polak, who was born in Torun, Poland. My grandfather changed the spelling of our name from "Pollak" to "Polák" in 1924, 5 years after Czech independence.
However, though this is an example of Polak as the name of a Jewish family, it is sometimes not so. The names Polák ("Pole"), Cech ("Czech") or Uher ("Hungarian" in older Czech) are as natural, after all, as Englishmen called "Scott", "English/Inglis" or "French" - all of which do occur.
Sorry that I don't know about Lillian Pollak. She might be related to my family, as many of my relatives lived in Olomouc, an eastern Czech city with good communications with Hungary.