How can you call yourself a muslim and be so disrespectful to your wife and children? Marriage is half the deen my brother. You gotta make this right.
Sit down and speak to your wife about it, pull out some hadiths to support your side and read some quran together. Assure her that you love her, that you think she's beautiful. Share with her the benefits of having a co-wife, like having someone who will homeschool the children if she decides she wants to work, someone to share the housework with, etc.
Don't continue with this haram behavior, because as long as you do, can you really call yourself a muslim? No, you can't. Polygamy in Islam is not about having secret wives, it's about a man caring for, loving, and respecting multiple women. You are not being respectful to a woman whose house you role through whenever you want.
As the man, you have complete control of this situation, don't try to peg it on the side piece. There is a reason why God made men the protectors of women. You are not doing your duty right now. If you really love this woman who you are disrespectfully referring to as "side piece", then there is no reason why you shouldn't ask her for marriage and begin living a halal life.
Until you are able to correct this behavior, please don't identify as muslim. We already have these extremist making us look bad enough.