If someone has their business registered in UK, states they are resident in Ireland. Do they pay their tax only in Ireland? Can they only work in the UK? what happens if they also work the odd day in Ireland for cash?
Very generally one should only pay tax in the country where you are tax resident. There are exceptions and this includes where the person has a tax base in the United Kingdom and this is called a permanent establishment. It is a simple question to answer. Cash is treated the same way as any other earnings.