無し is a classical style adjective meaning "no", "without", or "not existing" . and this is the predicate form, which will appear in a dictionary or can end a sentence with. Today this form create words , that mean "something without ....", Or " someone who dose not have ..." . On the other hand , 無き is the participle form of 無し. that means it will be followed by a word like Noun ,for exsample, and modificate the word. So usage could be 無し: 宿題は無し there is no homework。ゴールは無し there is no goal ( football game)。能無し someone who doesn't have any skill, in other words incompetent. 無き: 　国境無き医師団 Doctors without borders、仁義無き戦い battles without Honor and humanity. It is the title of series of yakuza movies. There might be my misunderstanding or typo, in this case please correct me. Hope it will help you.

On second thought, my interpretation may be wrong:( I don't know the whole lyrics of the song, so understanding the meaning of this part alone is very difficult. 「口無しkuchinashi」 may be a specific character and the character (who doesn't say a word or cannot speak at all) actually sings a song. Show the whole lyrics to your Japanese friend and ask him or her about the meaning of 「口無し」.

Oh, 「口無し」? Maybe that Japanese thought 「口無し」 is the name of a character like 「かおなしkaonashi」 in "Spirited Away" (「千と千尋の神隠し」). Actually 「口無し(kuchinashi)」 is not the name of a character; it is the name of a tree ("gardenia"). We call it 「口無し(literally meaning "without a mouth")」 because the fruit of this tree does not open even when it ripens. So "kuchinashi" has double meaning in Japanese: a person who does not say a single word and gardenia. So, 「口無しは唄う」 is a bit strange and contradictory sentence, but sounds very mysterious. That is, "A tree that is thought to be unable to sing because it has no mouth sings a song" (=> Japanese people would think "Oh, it's impossible, but I want to hear the song if it really sings."). This is my personal opinion, of course;)

Thank you. I just need some examples that could easily come to your mouth. The phrases that you would use in your own speech. I've got an example from my side. It's from lyrics. 「口無しは唄う。」One Japanese told me that 口無し sounds like a name of a character. Is that right?

Hmm, it's a bit tough question for me 'cause I don't know in what situation you want to use these words, but I'd use these two words like this: 「今、一文無しです。」(I'm broke now.) 「罪無き子供」(an innocent child) I hope these example sentences will help you:)