Adverbial participles are flexible little critters with a wide swatch of possible meanings. Usually their meaning is relatively clear, sometime very clear, but other times not so much.
In 2 Timothy 1 Paul is encouraging a discouraged Timothy. When I teach this chapter, I ask the students to discover every way Paul encourages Timothy. It’s a good exercise, and one that everyone in authority should practice. There is at least one unique encouragement in every verse.