I've said it a million times. Too much honesty kills a relationship. You say you didn't want to get married with a secret between you, so emptied your guilty conscience onto your bf... In my view, that was incredibly selfish, all it did was hurt him, what were you hoping to achieve? That he chalk it up to 1 more mistake? No, it was a betrayal, and those hurt like hell. If it was relevent to the current situation, eg your bf needed to get tested for a STD, then, yes obviously tell him, but a kiss with his cousin from years ago???? That just seems a cruel thing to share and I assure you that a confessional with a priest would have been much better for you both.

There's honestly nothing you can say to make him believe you are now trustworthy. Trust is earned, not freely given, especially not to someone who's broken it repeatedly in the past. And while he won't give you an opportunity to show him you deserve his trust, your hands are unfortunately tied.

Maybe in time he'll once again decide he can't live without you, maybe he'll give you a chance to prove you've changed. In the mean time, I suggest you cut your losses and move on, give him time to heal from this wound. And please learn from this experience that their really is such a thing as too much honesty. Damaging information of long ago, that have no impact on the current relationship should definitly be left to decay in the closet.

Best of luck