No, it's not allowed to work under RP at 2nd job!

I'd say, Z visa/RP sponsor can have an objections to your part-time job.

I did the same at my English teaching in China and elsewhere, butT ...

I casually mentioned at my prime job, some kindergarten or another school asked me to work with them part-time. Then, my Q: "Is that permitted?"

Answer from RP sponsor was always "Yes, but you can't be late or miss the classes at our school ..." with my reply: "No, our work schedule has a priority, and I'll arrange classes at kindergarten only in my free time."

When I cleared that, I undertake any extra teaching hours at other schools and private students in my free time.

Sometimes, teachers at my prime job asked me if I'm willing to have some extra classes elsewhere.

I accepted after the talk with School's principal.

I suggest, you test the felling at your Z/RP sponsor and once you see they don't object, you can work at 2nd job.

Keeping your 2nd job as a secret from your employer won't work, 'cause you're laowai and Chinese know exactly what you do in your free time.

However, despite your employer's agreement for extra work, you are still in violation of Chinese Labour law, and even if your sponsor agrees to your extra work, you can still get in trouble, because it's clearly written (somewhere ... ) that under Z/WP, one can work only at sponsor and nowhere else.

Penalties ... I'd say, there won't be any warnings and you'll be required to exit China in short exit time.

It never happened to me, so I can't really advice how is when manure hits the fan ...

To build off of what Icnif77 said, not only can you not legally have a second job, you also can only work at the location specified on your work permit. School/center has several locations around the city? Too bad, you can only work at one. 
Penalties include: fines, being arrested and detained, deportation and banned from entering for a number of years, and difficulty securing a visa/permit renewal denial.