Reporting part-time earnings while on benefits - 13 weeks in

I picked up part-time shifts as a janitorial supervisor and I am confused about how to report them. I have read a lot of threads here and could not find my exact situation. Sometimes my paycheck is late by a week. Who can I ask without risking my claim?

The claimant handbook has worked examples for part-time pay. Mine was clearer than the phone line.