Reporting part-time earnings while on benefits, graphic designer here

I picked up part-time shifts as a graphic designer and I am confused about how to report them. There are 3 people counting on me, so I am trying to be careful. Sometimes my paycheck is late by a week. Who can I ask without risking my claim?

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.