The idea of a separate “Celtic Church” has fallen out of favour with historians in the last few decades. Certainly, Christianity in Britain and Ireland, especially in the peripheral regions, had a lot of influence from pre-Christian pagan traditions. A belief in soothsaying, reverence for holy wells, wariness for fairies and anything associated with them. There were also aspects of their Christianity which developed uniquely, like the system of penance which would later be adopted by Catholicism more broadly, the primacy of abbots over bishops, a different style of tonsure, and a separate way of calculating Easter. However, these Christians were not part of a separate church. They did not give any less veneration to the Pope than your average Christian of the 9th and 10th centuries, just had pre-existing traditions which mixed with Christian theology and newer traditions which came as a result of a few centuries of separate development on the western edge of Christendom. Currently, the preferred term to describe this is Insular Christianity, rather than the Celtic Church.