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When you reconciled did you have a marriage contract signed?
Are there other assets that were brought into the relationship or acquired during the relationship?
The problem that you are going to have is that you added his name to the title. So essentially you both own the house.
It is possible that you can get a court order allowing you to buy him out. In terms of what you would pay he would be entitled to share in the increase of value to the house from the time you reconciled to the time you separated.
On top of that, if he contributed more to the value of the house than you did he may be able to get a larger interest than just the share in the increase of the value of the house.
I suggest that as a next step you consult with a family lawyer face to face. You can certainly do this without letting your husband know that this is going on.
As well, I suggest you take advantage of the fact that he is away and collect and make copies of any financial information that you can get your hands on so that you can provide that to your lawyer as well. In particular, if he is spending family assets on this woman and child it would be good for you to be able to prove that because then he is dissipating family assets and this will make your case even stronger in terms of your getting the house.