Yes, John Majors had an affair with another MP, because he had an affair with Edwina Currie. Currie wrote about it in her diaries. Apparently the affair went on for something like four years. She has enough details of the affair, and he has not denied it, that she can be believed.
According to most sources, John Majors did have an affair with another MP. There's no concrete proof, but most people say the evidence suggests that he had this inappropriate relationship. It's hard to trust the guy after what he's done in other aspects of his life, so he probably had an affair.
First I would like to point out that it is vitally important to indicate who you are actually talking about. There are two people with similar names: Johnny Majors (American Football) and John Major (Politician). Note the lack of an 's' at the end of the second name, this is actually pretty important. We are referring to the later who has been dealing with extramarital affairs. Personally, I can read that he has confirmed some affairs, but personally I couldn't care less what he did with other people. That's a private situation that he should have to deal with a home, not in public media.
I do not believe that John Majors had an affair with another MP. There is not enough evidence to suggest so . And unless there are more concrete proof of it, then people should not just assume it. I think that the issue is being manipulated for political reason by certain groups.
John Major's did not have an affair with another MP. No one is that stupid to have an affair with another MP, especially someone who is already being looked into. If he was stupid enough to have an affair with another MP, he would have been caught and everyone would know about it.