Yes!! This is so frustrating as an adult student!
I continue to formally study guitar and am also a teacher of guitar. When I'm in "student" role, I like to arrive at the studio venue early and sit down with my guitar for some pre-lesson warm up time. I usually go through some technical exercises for left and right hand. If there is time, I play through a few of my problem passages and think about what I need help with at this lesson. My warm up area is usually semi-public so if I can work through the anxiety during my pre-lesson warm up time, I'm much better prepared to learn when it's time for my lesson.
At this point in my studies, my instructor expects me to be ready to jump into what I am studying--but only because I have been taught and have learned strategies for dealing with my performance anxiety (which is what our "horrible playing" at lessons boils down to). My first run through of a piece is still often a bit rough but I can usually keep myself on track--not something I've always been able to do. The classical teachers that I've studied with over these 13 years that I've been playing have been quite gracious with me and provided support as I've worked through my performance anxiety.
As a teacher, I find that young kids don't often have this problem but once they hit pre-teen years and into adulthood, they become so much more self conscious. I almost always begin lessons with technical warm up so that we can work through the anxiety, address concerns with positioning and correct right/left hand technique on a more neutral canvas. It is less threatening. Then we begin the work on repertoire selections. Depending on the student and what we are working on, I may give them very clear parameters...such as "could you play the first 4 measures for me?" I work through various sections with them, praise them for their gains, give suggestions to improve and set goals for the next lesson. When all of this is done, THEN I ask them to play the entire piece. They usually end with a smile on their face and hopefully a sense of accomplishment and clear direction of what they need to work on to advance the piece. By taking this approach, I'm also teaching them how to practice at home.
The only time that I ask them to play a piece in performance mode at the beginning of a lesson is if we are nearing a performance date. In this case, I warn them that I am putting pressure on them in preparation for the upcoming performance. I encourage them to do their best, play through their mistakes, work their way out of memory lapses and no matter what, don't stop. They usually do better than they expect!
So much music, so little time.