Rav Moshe wrote a teshuva supposedly about Carlebach, although he does not mention his name. It’s “understood” who Rav Moshe was talking about. Teshuvot Iggerot Moshe, Even Ha-Ezer, I, no. 96
I don’t have it in front of me, nor can I find it on Hebrewbook.org.
Perhaps someone can help.