No — you can only push your solution once.
You're welcome to commit as much as you want locally. However, you only have one opportunity to push your solution. Once you push, the company receives notification and may begin reviewing your work at any point.
Accidental Push?
If you accidentally push your solution before it's ready, contact the company and request a new invitation. You can then add the new Git repository as an additional remote to your existing local repository, allowing you to push your completed work.