Paul speaks of Timothy, who served Christ with Paul, as if he were a child serving his father. Paul is emphasizing the close father-son relationship he has with Timothy in serving Christ. (See: [[rc://en/ta/man/translate/figs-simile]])
Here "the gospel" stands for the activity of telling people about Jesus. AT: "in telling people about the gospel" (See: [[rc://en/ta/man/translate/figs-metonymy]])