Background is that I'm an academic. I'm currently signed off sick from work due to bipolar depression. I have some work to do for a different university in a couple of weeks. They pay me. I feel able to do this work. It's easy and not too stressing. It involves travelling to another university for a couple of meetings. However, I don't feel quite ready to be back at my full-time/main job. It's very stressful right now and the type of work is just too much for my poor brain to cope with.
So my question is: can I be signed off sick from my main job, but do this small job for someone else? The chances of my main employer even knowing this is minuscule and they would probably appreciate the differences. I'm hoping to be maybe need an extra week so it's not a huge difference. But I don't want to break any rules.