a third time: Paul's first visit was when he first preached in Corinth (Acts 18:1–18). The second was an unexpected "painful visit" (2 Corinthians 2:1). Paul hopes the next visit will be happier, but there is still some distrust. They need to clear the air.
I will not be a burden: Paul did not want their money. He wanted them to fully accept his authority. He was willing to give his time and energy for them instead of taking payment.