This is a part of clarinet playing I find really difficult. I understand why it works the way it does but I have no idea how people do it so easily. Playing the high A is hard enough, I have that now though it still needs work, but then playing the next notes up, B, C etc, is really tricky and it stops me wanting to play clarinet unless I just play the chalumeau range. But I like jazz clarinet, and that needs the high notes.
Joseph Sellner’s book Studies for Oboe is chock full of short exercises that are great for working on the break on clarinet. Also, keep your left index finger close to the keys and don’t be afraid to use a tiny bit of left wrist action when pressing the A key, like turning a door knob clockwise. Just a bit though.
Practice slowly and make sure that you place all of your fingers very accurately in the same place on each tone hole every time you close a key, otherwise you’ll get tiny leaks and notes will fail to speak. Feeling of proper finger placement is a big part of clarinet playing. The more closed keys, as with B and C, the more chance you have to not properly cover a tone hole. It is also common to close several right hand tone holes while playing in the break in anticipation of going into the clarion register (B, C, etc.) That’s a bigger topic, however.