Permitted Work query

I am not sure if this question is relevant to Permitted Work, but it may be. I have read the booklet and have done a search but could not quite find what I was looking for.

Is it possible to claim ESA if you are already doing a little work, but it would only be a few hours worth maybe because your health did not enable you to do more.

For example you might have a job of only, say 6 hrs per week which you want to do but it is not enough to live on, but it is all you are able to do because of pain, fatigue etc.

If you then put in a claim for ESA how would that be regarded? Would they perhaps accept your application for ESA and then consider it Permitted Work if you 'passed' the medical assessment, or more likely would they just decide you were fit for work?

Are you in fact permitted at all to put in a claim for ESA if you are doing ANY work,even if only a few hours worth due to your health?

My son has a significant frontal lobe injury (head injury and meningitis). At present he is doing supported permitted work trying to 'make it' as a self employed artist (no income to date). He has been awarded middle level care and lower getting around DLA indefinitely and has just come up for ESA assessment. He is also significantly dyslexic and has dyscalculia He has night into day,problems with social interactions when he looses his temper, difficulty processing information (processing speed at 3rd centile - causes profound fatigue to the point of collapse) (in many ways he is like someone with autism), anxiety and mild agoraphobia. This means that some sort of self employment like this is his only option; he would not be able to cope in a 'standard' work environment. How will this impact on his ESA assessment?