I'm married and my partner is non binary and poly and I knew all this going into the relationship but they just recently decided to start dating. I was on board and told them to do it, but they invited someone over that night and did sexual acts in our place on our bed. I said ok don't do that again, but then the next night I wake up to them kissing and moaning on the couch as I slept (I work nights and sleep during the days) and I said to do it at their new partners place. Two days after that they say they are going to the next town over on a date and I said ok let me know what happens and later that night I find out from them they decided not to go and just went to their partner's place and had sex and didn't come home until the next morning. I don't know what to do because I know they have a mental problem with communication but I'm just In so much pain but I still wanna forgive them.
First off, as much as I am very hesitant on this blog to suggest that the partner who's less comfortable with a polyamorous arrangement should make "rules" to try and keep themselves feeling safer and more okay with it, this is a huge exception to my belief. Asking them not to have sex with a new partner WHILE YOU ARE IN THE HOUSE is completely reasonable. The timeline that you laid out here makes me question their judgment - that within 24 hours of you giving your blessing to try opening the relationship, they not only went out on a sexual-romantic date, but they brought the person back to your place and had sex with them.
Your partner is asking something very big from you - to be okay with them dating other people, even though that isn't your explicitly stated relationship orientation or preference. Usually, when you ask someone for something that takes lots of effort on their part, you should go out of your way to make it easier on them, not harder. I wrote that your partner was being 'selfish,' but then I realized that isn't even accurate - it's not remotely in their best interests to make your introduction to a polyamorous arrangement so painful and intense. I'm just baffled as to what they're thinking. If I was really invested in my partner being okay with something, I'd go out of my way to make it as easy as possible for them.
You say that your partner has "a mental problem with communication," but that's an explanation, not an excuse. If someone has a diagnosis, past trauma, or just a personality trait that makes communication harder for them, that means they need to use that knowledge as a tool to help them find ways to solve the problem - not just opt out of healthy communication and abdicate their obligations in relationships. If they know they have issues with communication, that means they need to try harder, try different things, be even more sensitive to potential issues - not just demand that you be okay with everything they do because hey, they can't help it!
You are well within your rights to sit down with your partner and demand that this gets worked out. You can say "I know communication is tough for you, so maybe we could write emails back and forth rather than talking face to face, or sit down with a couples therapist who specializes in open relationships, etc." - but what you should not do is just cede to their refusal to engage on this. Then, let them know that the way things are going is absolutely not okay with you. Explain that you have some (very reasonable!) boundaries about truthfulness and privacy. Be clear that you will not be doing 100% of the emotional labor it takes to make this new arrangement work, and if they're not willing to meet you halfway and consider your feelings, either the relationship itself or, at the very least, the polyamory, simply will not work.