' to have a cuppa ' is to have a cup of tea. It is very colloquial and not what I would expect to see in an international language learning programme. I would never use this phrase myself and have only heard it on T.V. used by people from certain parts of England. To mark ' Please may I have a cup of tea ' as wrong as a translation of this sentence is hard to understand.
They seem to have changed it now so that "a cup of tea" is the preferred translation. I can see why they did it as "baned" is used as a colloquial shorthand in the same way "cuppa" is in English, and I am sure most Welsh people are familiar with that word so it might be helpful to connect the two. However I do agree that "cup of tea" should be accepted as well.
I think that paned is a colloquial form for cwpanaid "cupful", which is derived from cwpan "cup".
So you're not asking for a(n empty) cup, but for a cupful of something, i.e. a quantity of something liquid.
So this is using "cup" in "a cup of tea" in its meaning "a cup with its contents" or "the quantity contained in a cup" (= cupful).