If the tradesman is VAT registered but not charging you VAT in return for a cash payment, that is a serious matter for them.
To claim your expenses against tax you must have a receipt. I do not think that there is any legal definition of receipt.
I had just this issue with a Revenue audit. They questioned a receipt which although quite detailed in terms of work done did not have any suppliers details except the name. I was able to supply Revenue with his telephone number and they were satisfied with that. Probably because the vendor checked out.
I suggest that to cover yourself you need the suppliers name and pps number. No reason you cannot write these details on the receipt yourself.