How do you and the Church handle the Popes who were blatantly corrupt and clearly cared more about political gain than religion? (Alexander VI for example.)
There are a few points that could be made about this:
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God can do good through imperfect, and even downright reprehensible people. We are all human and imperfect to varying degrees, and no one is too evil for God to use. (Whether this applies really has to be decided on a case-by-case basis, as it is undeniable that some popes did cause net harm to the Church.) To use Alexander VI as an example, he brought a lot of order to Rome and protected Spanish Jews, among other things. It wasn’t as if he was wholly bad.
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The pope is not infallible in his personal actions. Thus, Church teaching is in no way undermined by bad popes.
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The fact that none of the so-called “bad popes” actually changed doctrine can be said to be strong evidence of the guidance of the Holy Spirit rather than the election of the bad pope in the first place being seen as the lack of divine guidance. (Anything you might think is a counterexample of this is best discussed in isolation, as each case is unique.)
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Tying in with point #1, sometimes the political actions of bad popes benefited the Church in some ways. For instance, Pope John XII was responsible for crowing Otto I as Holy Roman Emperor which, along with resulting in a more peaceful situation in Europe, eventually resulted in the pope’s own family losing their negative influence in Rome.
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It can be equally harmful to have a pope who is deeply religious, but a terrible administrator/diplomat. While it is ideal to have a pope who excels in both areas, and holiness is probably the most important criterion, bad popes seem to have been pretty good on the political field, which was necessary in the environment of Medieval and Renaissance Europe, where religion and politics were dangerously intertwined.
There is probably more that could be said, but I’ll leave you with that for now!