Can I work part-time on a D-4 visa?

I'm currently on a D-4 (language training) visa and I've been in Korea for about 6 months now. My Korean has improved a lot and I want to start working part-time to help with living expenses.

I've read online that D-4 visa holders can work up to 20 hours per week, but I'm not sure about the exact process. Do I need to apply for a work permit separately? And what kind of jobs are allowed?

Also, has anyone had experience with the immigration office for this? How long does the approval process take? I want to make sure I do everything legally.