I'm on the other side of this.
I recently found out my husband was having an emotional affair with a woman he works with after I checked his emails.
He reckons we weren't happy, and I wasn't showing him any affection. That was partly true, but the reason I wasn't showing him much affection was due to a combination of a very tough pregnancy, then a newborn with all that entails (no sleep, bleeding nipples trying to breastfeed, hormones etc, as a mum yourself I'm sure you know what I mean), and most importantly the fact that affection wasn't wanted. Every time I tried I was shrugged away and he didn't care.
When I found out about him and his colleague my world was turned upside down. How was I supposed to reconcile the man I thought I was married to, the one who was my rock, the one who I thought I could rely on to be on my side, with a man who was willing to lie to my face when I tried to talk about our problems, to sit on a chair next to me on an evening and email her talking about his feelings for her.
I'm assuming the reason you aren't still talking to him is that his partner found out. Just so you know, when I found out he said it was all just words, not a thing he said to her meant anything, it just felt nice to have someone to talk to etc etc. He apparently hasn't spoken to her since, she put a sick note in for a month after I found out. Not decided what I'm going to do yet, I'm on maternity leave so reduced pay and with everything going up I could do with some savings before I make my decision.
I'm assuming your affair partner has said similar things to his girlfriend to try and stop her kicking him out. Either way, it's rough wanting someone you can't have, but you are both in relationships so you should never have gone there in the first place (either of you). You've nobody to blame but yourself for how you are feeling.