The issue of the emails can be overlooked if, and only if, you are able to overlook them. Maybe you can, maybe you can't.
From a strategic view, it's not the most important battle to fight.
However, your husband's behavior may be so toxic to you that seeing him email her may send you into an emotional tail-spin for several hours. And that's several hours of your life being angry or upset, which doesn't help your divorce-busting efforts.
Being needy, angry, clingy, chasing, etc. certainly will push him away. Trying to end the relationship now will certainly push him towards her.
But asking him not to carry on the affair in front of you, out of consideration, is not asking to end the affair. It's not chasing him. It's laying down a boundary that may be necessary for your mental health.
It's theoretically possible to ignore the emailing. Maybe you can't. It's up to you. I'm more concerned with your mental health.
Remember, showing some strength, by setting a boundary, combined with "Getting a Life" may prove attractive. People, in general, aren't attracted to people who allow others to violate their boundaries.
If you are not asking him to end the emails in front of you because you are afraid of upsetting him, then he's already won. He has YOU walking on eggshells in your own home while he carries on an affair.
I think you should ask him one more time, in the language I suggested. If he doesn't then decide if you can ignore and detach enough to overlook it.
Treat him like he's an alcoholic. There's a fine line between being patient and enabling destructive behavior.
