Hello and welcome to JustAnswer. Please note: This is general information for educational purposes only and is not legal advice. No specific course of action is proposed herein, and no attorney-client relationship or privilege is formed by speaking to an expert on this site. By continuing, you confirm that you understand and agree to these terms.
I am sorry to hear about this situation. Under Under Connecticut case law, common-law marriages are not valid (see McAnerney v. McAnerney, 165 Conn. 277 (1973), also Boland v. Catalano (202 Conn. 333 (1987). As such, someone in your situation cannot claim any type of marital right.
Therefore, this is looked at from a civil point of view. If he is on the deed and you are not, then it is his property even if you are on the mortgage account. He can ask you to leave. You are considered what is known as "tenant at sufferance," who resides there at the pleasure of the landlord.
However, you can of course demand child support of the children are his.
Please note: If I tell you simply what you wish to hear, this would be unfair to you. I need to be honest with you and sometimes this means providing information that is not optimal. Negative ratings are reserved for experts who are rude or for erroneous information. Please rate me on the quality of my information; do not punish me for my honesty.
I hope this helps and clarifies. Please use the SEND or REPLY button to keep chatting, or please RATE when finished. You may always ask follow ups at no charge after rating. Kindly rate my answer as one of TOP THREE FACES/STARS and then SUBMIT, as this is how experts get credit for our time. Rating my answer the bottom two faces/stars (or failing to submit the rating) does not give me credit and reflects poorly on me, even if my answer is correct. I work very hard to formulate an informative and honest answer for you; please reciprocate my good faith with a positive rating.