First of all, that looks really good for your second time bare back! You should be proud
There is nothing "wrong" with what you're doing. You have a trainer/instructor which will always work into your advantage, you always have someone there to correct any mistakes you are making.
As for the balance issue and slipping to the left, you know you are doing it so try to counter balance it out. Relax as best you can and keep yourself upright and find and hold your centre of balance. I am no expert but this is what I did when I tried bare back.
Make sure you're distributing your weight evenly in both legs too because if you're putting more weight into the left one, that will cause your slip too.
Like I said, ya look great