What circumstances led mark being outside of pauls favor? how do we know this problem was resolved ? include scripture reference where appropriate

The young "John called Mark" is mentioned in Acts 13:5 as accompanying Paul and Barnabas. But he left them in the middle of the journey (13:13). 

Paul and Barnabas start planning another missionary trip in Acts 15:36-41, and Barnabas wants to take Mark again. (This is where we find out the "John" of chapter 13 is Mark.) Paul doesn't want someone who left them the last time, so they separate; Paul goes off with Silas and Barnabas takes Mark with him. 

The epistle to the Colossians, attributed to Paul, includes a commendation of Mark in 4:10, indicating the earlier contention was resolved.

Hope I helped,
Ms.Weasley