The answers are as follows: the building is the priest’s house (or manse) at the Catholic parish church St Mary’s (Church of Our Lady of the Assumption), and I was standing in Ffynnongroew Road (Ffordd Ffynnongroyw).
The other picture shows the building as seen from the direction of Wellington Road. The house looks spacious but is often teeming with people and is a meeting place for community groups. The present incumbent is Father Charles Ramsay.
First correct answers came from Mr Berwyn Jones of Rhyl. A non-Catholic! A Methodist, yet!
Well done, Berwyn.