I recently received an email asking me for a response to a popular Catholic argument for the papacy. Since First Clement suggests that the Corinthian church in the late first century was seeking help from the bishop of Rome in settling a dispute, instead of seeking help from the apostle John, doesn't that suggest that the Roman bishop had more authority than John? Or, if help wasn't sought, but, instead, the bishop of Rome took the initiative to involve himself, doesn't that suggest some sort of claim of authority over the apostle John, who was geographically closer to Corinth? Here's a thread that addresses First Clement and the papacy.
One of the answers we can provide to Catholics who bring this issue up is to ask them if they apply the same reasoning to the Corinthian church's efforts to seek Paul's help rather than Peter's earlier in the same century (e.g., 1 and 2 Corinthians). See the thread linked above for some other responses.