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In R' Wiesenthal's Siddur Sefas Yisroel, he talks about the כף הכינוי and different opinions about it. But I have no idea what this actually is. So, what is
This is the suffixed ך used to indicate a second-person direct object or possessor. Thus it's a כף (the letter of the alphabet) of כינוי (i.e. referring to someone or something).