Welcome to SI sissil898. You've come to a great place to find some help and support. Whether or not you can save your relationship is something no one here can tell you unfortunately. Infidelity is often a deal-breaker for many people.
What I CAN tell you is that there is no point in trying IF you do not offer one. In other words, what about you is different now than you were just a few weeks ago that makes you a safer person to be with?
Let me put it to you this way. Suppose you are running errands when a mugger jumps out of the bushes, physically assaults you, steals your purse and runs off. A short time later, the mugger comes to you and says he's sorry and wants to be friends instead. He tells you he grew up in a rough neighborhood and attacking people and stealing from them is just how he's always been. But he feels bad for attacking you, and hopes you can be friends.
Would you want to be friends with this person? Because they literally just beat the snot out of you and stole your purse recently. Why the hell would you want to be friends with them? Are they still mugging people?
Now let's change the story a bit.
Let's say the mugger comes back to you and says,
"Look, I had no right to do that to you. You didn't deserve what I did to you and that's all on me. I want you to know that I've taken steps to try and make this right. First, here's a new purse to replace the one I stole. I went ahead and replaced the cash and put twice as much in as I stole. Don't worry, I found a legit job to make that money, I'm never stealing again. In fact, I joined a support group for muggers who are trying to quit so I can change my ways and stop being a mugger and live a more authentic life. I've arranged and already paid for a therapist for you and I will work three jobs if I need to in order to pay for that. I want to make you whole from the damage I caused you. In fact, I was able to track down several people I mugged now, and have done my best to try and make amends with every one of them. I dropped all my mugger friends, moved to a safer neighborhood, got involved in the community watch so that no one, not you, not anyone, has to feel unsafe running errands again. I'm going to be part of the solution. And for what it is worth, I realize that all of this doesn't take away the fact that I hurt you. I don't expect us to be friends, but I hope that one day, if I continue to work hard at being a better, safer person, that maybe you'll consider it."
Now would you at least be more likely to consider it?
On a personal note, there is one thing I'd like to impart to you. Bipolar disorder is a harsh disorder to deal with, I get that, and when people go manic, they do and say things they sometimes regret. That being said however, there is nothing specifically about being bipolar that "makes" a person do anything. Being bipolar doesn't cause a person to cheat, and that's really important to understand. You didn't cheat because you are bipolar. You cheated, AND you are bipolar. In the same way that growing up in a bad area didn't cause the mugger to mug you, being bipolar didn't cause you to cheat. The mugger mugged you because in his head, he decided that what he needed was more important than your right to safety and security. Lots of people grow up in bad neighborhoods and don't become muggers. These are personal choices.
When people hurt others and then pass it off with an excuse, it pisses off the vitcim. Why you think you did something wrong is irrelevent, it doesn't help your victim at all, and it removes any and all reason for the perp to own their actions and choices, and to change.
Telling your partner that you cheated because you were manic is basically washing your hands of the problem. So now what? Suppose he comes back? What happens when you go manic again? Because last time that happened, you cheated, right? So what's different now? How does he know you won't cheat the moment you go manic? He didn't expect that to happen last time, and yet it did.
Look, if you want your partner to even have a reason to come back, or more importantly, if you want any future relationships you have to be safer, then YOU NEED TO CHANGE FIRST. Figure out why you cheated and stop blaming the mania. The mania only helped let the unsafe urge take over. It didn't create the urge. So figure out why you thought it was okay to cheat at that moment. Take steps to be a healthier person.
TL/DR; Don't ask your partner to come back, give him reasons to. If you can't give him reasons to, then don't expect anything more.