Do I need another schedule C for my online ESL Chinese teaching job? Or do I put that income on my same schedule C as my music business income? If so how do I do it?

Yes, each different type of work requires its own Schedule C. If you were to have multiple payers for one type of work, you would not. Each type has its own unique expenses.

See the screen shot below. It may say "Add another line of work" instead.