Hello Lily, and I'm too sorry your comment has been missed but pleased you have posted.
Whether or not cheating on a partner/spouse can be overcome the first time may depend on the situation you're in and if it happened for some specific reason is a decision these two people need to decide on, however cheating once again for the second time is indicative on what may continue to happen, the trouble is you've just bought a house together, but there will be problems even if this didn't happen.
Emotional abuse in a relationship is not a pleasant way to live with someone, especially if you're supporting him financially only complicates the situation you're in, as well as trying to cope with his depression and if he has been unable to tell you the truth only makes this difficult for you to know what to do.
If he was remorseful the first time and getting help for his depression, then perhaps this could be overcome by talking with a counsellor together, however, there are times when the negatives outweigh the positives, and this is something you need to decide on.
Whether or not your relationship can recover is up to how you believe but I would be consulting with a lawyer on a no win no fee basis to talk about your options, as well as talking with your doctor.