UPDATE! I can tell you that the very next mobile update that we are actively working on will allow you to mark workouts as ‘complete’ from the mobile app! Hold tight! 
If you’re interested in trying the beta out, you can sign up here: Early Access to iOS Beta: Updated TrainerRoad (link to fill out early access form in Pete’s post).
We should also clarify that it’s iOS only at this point, but Android will be following shortly after!