If your boyfriend is genuinely praying and struggling with porn addiction and he is in the church, making use of the acraments, going retreat, etc. then unless there are other obvious mismatches in temparement, lifestyle etc, I would not necessarily advise you to break up with him.
I am in the same situation as you, but with a boyfrined who is farther along the continuum in terms of his addiction. He has stopped the porn but is still masturbating. We are, however, in a chaste relationship. I do pray for lot for him and it is difficult sometimes to hear him fall but I also know that he is genuinely fighting it.
What makes it easier for me, though, is that I myself, a woman is a former porn addict. Becuase of this, it makes it easier to forgive him and have compassion for him when becuase I know the struggles which he faces. We both maintain with a lot of dialogue and prayer this chaste relationship and he has confirmed to me that the graces we have obtained through having a chaste relationship with him as well as praying with and for him has helped him to diminish the incidences of falling back into sin and he gets better and better.In addition he is demonstrating a deeper and more mature understanding of what the marital embrace will be when we engage in it.
One of the sites that helped my dear bf greatly is Shelley Lubben’s site. She is a former pron star who testifies about how she was converted and got out of the prorn industry with the help of her dh, Garrett.
shelleylubben.com/
Another great site to help him is this one
pornnomore.com/Serenellians.htm
*As a side bar to the people who have an understanding of this as only a sin. Yes, porn addiction is a grave sin, a mortal one. But it is also an addiction and with all addictions, there is the fruit, the sin of porn addcition *
and ** there is also the roots. Until I did a lot of work on my past hurts (having a dad who was unfaithful to my Mom, low self esteem, absentee father, sexual abuse, etc), I made some headway with avoiding the sin but not with eradicating it. Only now that I have done a lot of inner healing work with my excellent spiritual director that I am now able to have long periods of complete chastity -no porn, no fornication, no masturbation,very few if any sexual fantasies - and be joyful and free. So in dealing with this issue one should not only depend on the outer things to prevent such a sin from happening ( prayer, fasting, accountability partners, internet filters) but also do any inner work in terms of deep inner wounds needing to be healed to effect lasting change and chastity and freedom