I pay monthly payments to child support for back pay. I need to renew my p.e3.r.c. card but it has been denied. Can this be reinstated

What is a p.e3.r.c. card?

(blue card) issued by the state of Illinois for securtiy workers etc

And the state agency has "denied" it, you say? Do you mean they refused to renew it"

What reason did they give? Ask them to back it up with a specific statute or regulation.

Is there a court arrangement to pay the back support,

For the state of Illinois, if you have not caught up or currently do not pay child support that you should be you cannot be issued most state licenses except for state ID, another example of licenses you might have problems obtaining are the fishing and hunting license, firearm license, etc.