Reporting part-time earnings while on benefits (St. Louis)

I picked up part-time shifts as an event coordinator and I am confused about how to report them. I do not usually post, so bear with me. Sometimes my paycheck is late by a week. Do you report when you earned it or when you were paid?

Thanks for writing this out. I went through something close to it last year, and it settled down once I kept a simple log of every call and form.