This is a horrible story to read. Your partner doesn't sound like a wonderful man to me. He has lied to you repeatedly, cheated both while you were pregnant and now again when you have a young child. His agreements to stop contain veiled threats about him stopping but it inevitably happening again unless you get okay with it. This is somebody who falls far short of wonderful.
You have children to think about. If you think it's possible for your partner to change? To stop lying whenever life doesn't go his way and to behave ethically toward those he loves.
If you don't, I would leave. Children pick up lots from how their parents behave and the last thing I would want is any children of mine growing up thinking that it is okay to lie and cheat to get what they want. Or to grow up thinking that being lied to and cheated on by those closest to them is a normal part of life and something that they should just get used to.