Know it’s not the Game
First thing's first: Clarify your goal. Decide what you ideally want to happen. For me, understanding is usually my goal, and that makes it easier to figure out what's wrong together. Other times, my goal is clarify why something is the way it is- whether it's feelings or actions or rules. Some things aren't up for discussion, and making those hard lines clear can be really important. Whatever it is, decide what you want to accomplish and what progress would mean.
Second, if you think you're friend is annoyed over a game, you're thinking in the wrong place. If it's truly over 'small things' it wouldn't be so hard to confront or talk about.
Ask yourself questions. Are they right? Are they being mean, unreasonable, rude? Are they doing it on purpose? Are they starting something or reacting to something? What about it is hurting or bothering me? And most importantly: where is it coming from? A big part of solving problems with other people is understanding what you're doing and feeling, so it's easier to tell people and talk about. The more prepared you come to the confrontation, the easier communicating and understanding each other will be.
Sometimes when it's about 'small things' it's a bunch of small stuff they dismissed and let slide. When you ask, make it clear you're listening. You're asking because you want to know something you don't know. You're not asking rhetorical questions (at least you shouldn't be ._.) and you're not trying to make them say something they don't mean. Be clear that your intentions are to resolve whatever conflict is there.
Appreciate what they tell you, whether you wanted to hear it or not. Always thank them for what they're willing to share. What you do with that information is all up to you. Communicate as gently as you can, and ask yourself how you can reach that goal you set at the beginning with the information you have now.
Take your time. Fixing and changing things always takes time. You're learning. Your friendships should always be worth the trouble
Good luck