Paul fears that his critics will seem be vindicated
If we consider the context of these verses, it's clear that Paul is extremely frustrated with the Corinthians. This will be his third visit and they still don't seem to understand certain basic principles. In 1 Corinthians, some are following Apollos, some Peter, some Paul. They are disorderly in church. Some of them are suing each other. Most important for this particular question is the penchant for immorality. Some thinks their freedom from the Law means they can go with prostitutes without endangering their salvation. One man is even having sex with his step mother. (1 Cor. 5-6)
Some of these issues are still a problem now, as Paul prepares for another visit. So he says, as the OP mentions: "I may mourn over many of those who have sinned in the past and not repented of the impurity, immorality and sensuality which they have practiced."
If these practices have continued, it would be particularly humiliating for Paul in the context of an immediate challenge to his authority from the "Judaizers," against whom he rails in chapter 11 and 12 in similar tones to his letter to the Galatians:
Are they Hebrews? So am I. Are they Israelites? So am I. Are they
descendants of Abraham? So am I. Are they servants of Christ? I am a
better one-—I am talking like a madman—with far greater labors, far
more imprisonments, with countless beatings, and often near death.
(11:22-23)
Paul goes on at length to let his audience know how truly disturbing it is to him that this church, which he considers to be his spiritual children, has been partly seduced by a false doctrine, and his authority is not fully accepted:
I have been a fool! You forced me to it, for I ought to have been
commended by you. For I was not at all inferior to these superlative
apostles, even though I am nothing. The signs of a true apostle were
performed among you in all patience, with signs and wonders and mighty
works. (2 Cor. 12:11-13)
Why would Paul be humiliated?
Let me propose: the reason that Paul would be particularly humiliated by continued sexual sin in the Corinthian church is that this type of immorality hit very close to home. It would tend to confirm his opponents' criticism of his doctrine. For Paul:
“All things are lawful for me,” but not all things are helpful. (1
Corinthians 6:12)
For Paul's critics, the moral commandments are not only "helpful," they are essential, especially the commandment against sexual immorality. His opponents must have harshly criticized Paul's idea that Christians are free from the requirements of Law but will naturally live moral lives as fruits of the spirit. Moreover, subjecting oneself to this particular moral law taught by Moses was not only a demand of the so-called Judaizers, it was agreed to by Paul at the Council of Jerusalem, according to Acts 15:20.
For the Corinthians to continue in immoral practices would cause Paul great humiliation, as it tended to undermine one of his basic teachings: that being "free in Christ" would naturally bear the fruits of the spirit in the form of a moral life. This humiliation would be especially painful for Paul, given the apparent challenge to his authority in Corinth by teachers who held that the Jewish prohibition against sexually immorality must be upheld in the Christian churches.**