Old English clut "lump of something," also "patch of cloth put over a hole to mend it," from Proto-Germanic *klutaz (source also of Old Norse klute "kerchief," Danish klud "rag, tatter," Frisian klut "lump," Dutch kluit "clod, lump"); perhaps related to clot (v.).
In later use "a handkerchief," also "a woman's sanitary napkin." Sense of "a blow" is from early 14c., from the verb. Slang sense of "personal influence" (especially in politics) is by 1946, American English, on the notion of "punch, force."