RobH_en_tn/php/02/22.md

as a child serves his father, so he served with me

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)

in the gospel

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)

I am confident in the Lord that I myself will also come soon

"I am sure, if it is the Lord's will, that I will also come soon"