A common complaint of many students is that they never play as well at their lesson as they do at home. They are frustrated but at the same time seem to accept this as a fact, as something that is just the way it is, nothing you can do about it.
Here are my thoughts on this matter.
The reason you don’t play as well at the lesson as you do at home is because you are not as well-prepared as you think you are. At home, you are comfortable and relaxed, and you probably never perform, instead you can always try again if something wasn’t quite right. At the lesson, you are nervous, and you expect to play your piece once and get it correct the first time.
If you want to play at the lesson (almost) as well as at home you have to practice differently: you have to be much more detailed in your approach: hands separately, not just once but many times, with metronome and without, looking at the book as well as from memory, ultra-extra slow, then faster, repeating one measure or phrase at a time, many times, always listening for things to improve (and then improving them), hands together with the same focus as HS, then stringing two measures or phrases together, etc and so on. That way you get to REALLY know the piece, or part of a piece, not just from having played it many many times, but from really understanding. Test yourself: before you start playing a section, look at the book, then look away and name the first note(s). Can you do it?
Since at the lesson your goal is to show me that you have improved or learned something new – show me by playing something, once – you need to practice this at home. With a completely different focus now, you have to set the goal to perform whatever you want to play/perform (whether it is two measures or two pages): start and keep going until you are finished. Do not stop and fix something, even if it wasn’t perfect. This requires a completely different mindset because you have to take mental notes but you cannot be bothered by mistakes. You need to practice to perform especially at the beginning of your practice session, when you are not warmed up yet (mentally, mostly) . The way you play then will be the way you play at your lesson.