My grandfather, Karel Polak, was born Karl Pollak in 1878 - in Rataje near Prague in what is now the west of the Czech Republic, but most of his family lived in Olomouc in Moravia (the eastern part of the CR). His father had been estate manager for the Austrian Count Auersperg, but on my great-grandfather's early death (from pneumonia caught touring the estate), Auersperg evicted his widow & her 5 children - my grandfather was forced at age 14 to learn a trade in order to provide for his mother & siblings. His Olomouc cousins helped, to the extent that my grandfather's brothers were able by the 1920s to build themselves a large Art Deco mansion in this small city. (With the restoration of democracy in Czechoslovakia/CR this has now come back into my family's possession, after it was confiscated by first the Nazis, then the Communists.)
The Olomouc cousins also endured some hard times, either late in the Austrian period (ended with WWI) or shortly after. 2 of them, Eduard & Wilhelm, emigrated to the US; I suspect that the actor/director Kevin Pollak (who is my age & resembles me a little physically) may be grandson of one or other. Some of their American descendants may well have re-spelled their surname as "Pollack" or "Polach", though "Polach" sounds more like the name of the Czech anti-communist martyr of 1968, Jan Palach (the "ch" is the "Bach" or "chutzpa" sound).
Nová Lhota is in the very picturesque Hornácko region ("HORR-nyaah-tzko" with short "o", the "n" has an accent like the Spanish tilde giving it this "ny" sound) of south-east Moravia near the Slovak border, in the foothills of the Carpathian mountains. Its country festival costumes & dances are also especially picturesque, & the area grows some fine wines.
The name Pollak/Polak comes from the Czech (& Polish!) for a Polish man. "Polák" with the right-leaning accent on the "a", is the correct Czech, but note that after the Austrian conquest of 1618, the Czech language was banned by the Austrians & it was more usual for the name to be spelled "Pollak" - phonetically for German, where the double "l" correctly makes the "o" a short sound. My grandfather changed the spelling back to Polák in 1924.
We are probably descended from Rabbi Ya'akov Polak, born in Torun (Poland) in the 1400s, named by the acronym "the SMA" for the Hebrew title of one of his books. He was a famous developer of dialectical method in studying Jewish scripture. To judge by the non-Jewish names "Markus" & "Hermann" of my great-great-great & great-great grandfathers, highly unusual for orthodox Jews in the late 1700s, the family became involved in the Sabbataean & Frankist heresies, & some members later became Catholics.
My grandfather, together with his brothers & sister, remained a Jew & a patriotic Czech; all were deported to Auschwitz or Majdanek extermination camps, & died there in 1943-4.
I hope that you found some of this interesting & possibly relevant!