How can I extend the license for CVX Professional Edition?

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I think you no longer need the professional or academic license version of CVX.
See Announcement: CVX 2.2.2, the first "full" open-source release .

As for Mosek license, that is a matter for you to address with Mosek, and not realy a CVX matter anymore.

Thank you!
I have read the announcement in the link (Academic license request | CVX Research, Inc.).
I deleted CVX according to the method in the link (Removing a CVX version completely) and then reinstalled CVX. However, academic licenses still exist. Do I need to reinstall CVX or continue using it directly after my academic license expires?

I think if you use CVX 2.2.2, the CVX academic license is irrelevant.

Mosek academic license is a different matter though, and is still relevant when using CVX 2.2.2.

Yes.
The installed CVX version is 2.2.2. (CVX: Matlab Software for Disciplined Convex Programming | CVX Research, Inc.). And I applied for an academic license on the Mosek official website. According to your suggestion, can I continue using CVX directly after the academic license expires?

After your CVX academic license expires, you can continue using CVX 2.2.2 with any free solver, or non-free solver for which you have a license. So you will still need a Mosek license in order to use Mosek as solver.

Thank you! I got it!