I have worked in the public sector my entire adult life.
The purpose of asking for all employment is mostly so that background checks can be made, if so desired, into all of your employment history.
If it is a federal job, I'd resubmit a new application.
If it is a state or local job, if you get an interview, just have a revised ready to take with you to the interview. If you know the HR office handling the application, then give them a call to see how to proceed.
If those two jobs are relevant to what you are applying for, then you may want that to be considered.