You can still leave the country, if you apologize for your fault and meet a kind immigration officer, you won't be fined.

In fact, now you can ask for help from your employer who can go to the Local Exit & Enter Department to apply for Residence Permit for you. Once you get the Residence Permit, you can enter and exit Mainland China for many times during its validity, now if you return to Singapore, you have to apply for a new China Work Visa(https://www.travelchinaguide.com/embassy/visa/work.htm) to go back to China, then apply for Residence Permit within 30 days after entry.