I. The Debate
Traditionally, a synagogue would appoint a pious Torah scholar as a chazzan to lead the congregation in prayers. Throughout the ages, the question has arisen what to do if a chazzan is found to be less than pious. For example, what if a chazzan is found to have stolen an object or to have had an affair with a married woman. The answer to such questions largely depends on the type of evidence for the wrongdoing which can include rumors of varying reliability, evidence that is indicative but insufficient for a conviction in a beis din (raglayim la-davar), or valid testimony. However, even when there is testimony from valid witnesses there is still an issue of whether to deal with the sin in a public way by deposing the chazzan.
The Gemara in Mo'ed Katan (17a) states that when a Torah scholar sins, the court should not punish him in public because that would serve as a denigration of Torah. Rather, they should "hide it like the night" and punish him in private. The Rosh (ad loc.) understands this to refer to any Torah scholar but adds that if this is leading to a Chillul Hashem, then they should disregard the admonition and punish him in public. The Tur (Yoreh De'ah 334) rules like his father, the Rosh.
However, the Rambam (Mishneh Torah, Hilkhos Talmud Torah 7:1) rules that this only refers to leading Torah scholars (chakham zaken be-chokhmah ve-khen nasi o av beis din). They may not be publicly punished but an average scholar may. But the Rambam adds that we should still not rush to publicly condemn even an average scholar. The Shulchan Arukh (Yoreh De'ah 334:42) rules like the Rambam but adds the Rosh's caveat that if any of this leads to a Chillul Hashem then we should punish the scholar in public. (The Sefas Emes [Mo'ed Katan, ad loc.] found this addition from the Rosh puzzling because the Shulchan Arukh rules like the Rambam and against the Rosh.)
Yet, the Rambam seems to contradict himself. In a responsum (Pe'er Ha-Dor 85) on the issue of whether to depose a chazzan, the Rambam applies the rule of the above Gemara to every public official and not just the leading Torah scholars. The Radbaz (Responsa, vol. 4 no. 809) resolves this contradiction by positing that when the official sins in private then the admonition applies to everyone but when he sins in public then it only applies to the leading scholars. This would seem to mitigate the Shulchan Arukh's addition from the Rosh, unless we assume that he could sin in public without it causing a tremendous Chillul Hashem. However, this distinction of the Radbaz based on the Rambam's responsum does not seem to have been adopted by later authorities.
II. The Conclusions
The upshot of all this is that if the chazzan's sins are proven then he may be removed from service, and so the Shulchan Arukh (Orach Chaim 53:25) rules. The Rema adds that we do not remove a chazzan simply due to mere rumors but we do remove him if there are witnesses or something like that (ve-cha-yotzeh ba-zeh). This last phrase means we may remove him even if there is evidence that is insufficient to obtain a conviction in beis din. The Mishnah Berurah (ad loc. 78 and in Bi'ur Halakhah sv. Im) rules that even strong rumors are sufficient to depose a chazzan.
Significantly, the Radbaz (ibid.) adds the following important comment when discussing the apparent contradiction in the Rambam's writings:
ואף על פי שהרב ז"ל מהדר אזכותא דתלמיד חכם שאין מורידין אדם מגדולתו ומינויו מכל מקום מודה הוא שאם עבר עבירה בענין מינויו כגון שהיה חזן לנשים ועבר עבירה עם אחת מהן מעבירין אותו ממינויו שלא לתת מכשול לפניו עד שיתברר שעשה תשובה שלימה.
And eventhough the Rambam tries to find merit for a Torah scholar, that we do not depose him from his lofty position, he would still agree that if [the scholar] committed a sin relating to his position -- for example, if he was a chazzan for women and committed a sin with one of them -- we remove him from his position so as not to place a stumbling block before him, until it is clear that he repented completely.This would presumably apply even more when dealing with a man in a position to force himself on others.
Additionally, R. Ya'akov Emden, in his glosses to Mo'ed Katan (17a), makes the point that certain sins are punished publicly if there can be no presumption that the scholar repented for them (I hope that I am reading this correctly). His example is heresy, about which it says "None that go unto her return again" (Prov. 2:19). This was certainly relevant to him due to his concerns regarding rabbis with hidden Sabbatean sympathies. Perhaps the same would be true about a scholar who commits a sin about which we know statistically that it is unlikely for someone to stop committing it even with extensive treatment, and certainly not without.
(I should add the obvious caveat that any real life situations must be brought before a competent rabbi or beis din.)