I think what you're feeling is entirely reasonable in your circumstances. And R, too. I've been the the primary when a new secondary came into the picture and what both of you are feeling is normal. The fact that you can see H only once a month probably means the "new" will take a lot longer to wear off and I can understand R's feeling a little left out about the intensity of your feelings for H. If R is like I was, he's going to need a lot of reassurance that he's the anchor in your emotional life and you're afraid of hurting that.
Nothing does it like keeping talking, and that means baring your souls to each other about your hopes and fears of your relationship. It doesn't sound like there's much overt conflict about what you both want so I think you should be able to work things out so you stay together. (I'm picking up that's what you're really worried about). Reaching an agreement, some call it ground rules, isn't just about physical stuff, it's about emotional needs as well and is an ongoing process if you know what I mean. Sort of like 'I want you emotionally available to me when I need you and if I say these words "------", it's important and I need your time. I can share your love but I need my share, too."
All relationships are complicated, and it's just more complicated when more than one lover is involved so it's reasonable to have the feelings you're having.