When the Greek is ambiguous, it is always nice to maintain that ambiguity in English. But usually you can’t and you have to make a decision. However, in this verse the ambiguity is essential to the meaning of both the verse and the passage.
2 Tim 1:3–5 is one long Greek sentence, and it illustrates the dangers of breaking up a long Greek sentence into shorter English sentences, and also the danger of connecting the phrases merely sequentially. Why was Paul so thankful, despite the fact of his imprisonment and coming execution?
When Greek lays out a sequence of events, it tends to use anarthrous participles followed by an indicative verb. The question for the student is how to translate the participle into proper English. In our passage, the participle is actually telling us why the false teachers turned aside to error.
This is a rather technical blog on the nature of Greek appositional constructions. I imagine I lost many of you with that first sentence, but the distinction between the two forms is important exegetically. If you read "x of y," is "y" the same thing as "x" or is "y" explaining something about "x"?