Enemies deal damage to your lid while it’s closed. Try to defend yourself during the moments when the lid is shut.
Also you can upgrade a skill that strengthens the lid. It’s in the second row of the second branch. If you're really struggling to protect the lid, try replaying the game and prioritizing that skill first.
However, the game is balanced to give you a bit of time, so it is possible to beat the boss even without that skill.