Make yourself sure
When you meet X after the breakup, make it clear in front of him that you have thought a lot about your relationship and are ready to improve yourself. Admitting your mistakes, say that you know the reasons why both of you had a breakup. Also, assure that you will not repeat those mistakes in the future. Like - you should have told that day that you have a trip plan with friends or why you were angry with me coming late, I could not take care of you.
Maintain distance
After the breakup, in the same way, whether he is angry or in trouble, it is on you that he is being irritated constantly by phone or message. Hey, your relationship can get worse in such a situation. So let them live alone for a few days and you too live your life.
You must have also heard that if your love is true, it will definitely come back to you. You feel that you are persuading them by calling or messaging, but it may cause them trouble and irritate them rather than obey them.
https://www.youtube.com/watch?v=Sdsh2MKZai4&t=1s
Do not make the mistake of showing favor
When two people are in a relationship, they support each other in happiness and sorrow. Some do more and some less. Maybe in some bad times, you have also supported them. Even if you have a breakup, these things, these moments will be memories for both of you. So do not try to show that you have done a lot of favor to them, because love is just love. Here there is a place for care, support, not favor. So if you want to get them in return for your favor, then this dream will remain incomplete. It may be that he will get even further away from you on hearing these things.
Proceed with the plan
Your X is ready to give you one more chance, so make a plan first so that your relationship can become stronger than before. First of all, understand what the two of you expect from this relationship. For this, you can ask X what are the expectations she has from this relationship, what does she want from you, which you fail to fulfill first.