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."
"to beat, strike with the hand," early 14c., from clout (n.), perhaps on the notion of hitting someone with a lump of something, or from the "patch of cloth" sense of that word (compare clout (v.) "to patch, mend," mid-14c.). Related: Clouted; clouting.