What does 1 Corinthians 4:21 mean?

Paul sounds especially "parental" in this verse. That fits, since he has described himself as a father and the Christians in Corinth as his beloved children (1 Corinthians 4:15). The problem is that these little children (1 Corinthians 3:1–2) have been misbehaving. He is writing to correct their wrong thinking, bad attitudes, and careless behavior.

Some in Corinth, though, will continue to resist Paul's instruction to them. Paul knows this. He has called them arrogant (1 Corinthians 4:18). In their pride, they have decided they can follow Christ in their own way. They don't have to take Paul's teaching as the final authority about what is true or not. Paul fully expects some to read his letter, and proudly insist that they have nothing to change, or no need to hear Paul's words.

Paul has warned these arrogant people that he is coming to see them and that he will come once again backed by the supernatural power of God, displayed through the Holy Spirit. His readers likely remember very well seeing God's power demonstrated in this way (1 Corinthians 2:1–5).

With that in mind, and very much like a father, Paul asks if they would rather he show up with a rod or with a loving spirit of gentleness. Paul is using the father/child metaphor to make a point. He is not demanding to be called by the title of "father," or asking for other honors. He would not bring an actual rod, which in that era was indeed used to discipline children. However, God may choose to use His power through Paul to bring correction or demonstration of power to those living in rebellion.

Of course, Paul will come with love for them in either case. Even discipline would be a display of God's love for the Corinthians (Hebrews 12:6). Or that love can be expressed in a spirit of gentleness, as Paul encourages the Corinthians and helps them to change course in how they live out the truth of the gospel.