I have to agree with the above post. I'm going through a situation now where my wife is and has been seeing someone else. It may not seem terribly healthy to have this attitude but if she was to decide to work on our relationship the two things that would help me regain trust are:
1) Being completely straight about any questions I have about the affair. Personally, I want to know how long it has been going on, how far it has gone, etc.
2) When she goes out, I would like to know who she is with and when she is going to be back. A phone call would be nice if plans change.
Now, on my end, the need for #1 and #2 has to be tempered somewhat. Can't take it to extremes...at least not once some progress is evident.
Don't mean to turn this into a post about our relationship but I think there are some parallels between your situation and ours and I'm sort of in the same seat as your H.