"Czyk" is associated with Polish royality back in the 15th century, under the reign of Kazimierz IV
There is not an iota of truth in this statement.
The suffix -CZYK signifies a Polish version of a son of somebody, such as in RYBACZYK (son of RYBAK, a fisherman), KOWALCZYK (son of KOWAL, a smith). The Ukrainian version would use the suffix -CZUK, Americanized as -CHUCK .
On the other hand, the prefix BEZ- means "without" in Polish.
but I cannot decipher the rest of the last name.
BEZRUDCZYK:
It is very probable that the surname BEZRUDCZYK is connected to the word RUDA, an ORE, except that in the Old Polish that word had several meanings. It mainly meant the same thing which it has now - "an ore, or specifically and iron ore from which iron is melted.'" The second meaning is "a red soil" and the last one means "a swamp". Those three names are not mutually exclusive since it is well known that iron can be found in so-called RUDA DARNIOWA (turf iron ore) and RUDA BAGIENNA (swamp iron ore). The adjective RUDY/RUDA also means a red colour, as it can be seen anywhere where iron is present. And finally, a noun RUDY/RUDA also means a redhead (male/female).
So for example, if an early iron ore prospector, who often came back empty handed, only with samples of barren rocks, BEZ RUDY (with no ore) he could be teased as BEZRUDY.
His son could then inherit the nick BEZRUDCZYK, after his father's nick.
Alternative explanation:
If by any chance BEZRUDCZYK is a corrupted version of BEZRODCZYK then its etymology dramatically changes. The root of the words becomes RÓD (Pan-Slavic ROD), meaning an extended family, a clan, a tribe. From this a family in a narrow sense could be derived: Polish RODZINA, Pan-Slavic RODINA.
Consequently BEZ-ROD-CZYK would mean a son of somebody without family, without parents, orphaned.