To put it briefly, he just wants them to have an open and honest relationship, hence genuine.
The longer version :
Hachiman is basically fed up with the facades that happens in Service Club to maintain their "friendship" that became more fragile in the time since Tobe's confession request, Iroha's election, and Christmas event.
After Tobe's request, the relationship between all of Service Club's members becoming more distant, but they are still trying to be together because they all want to cling to this "friendship". The reason for maintaining this status quo is because they've never had relationships this close with other people before. Having a taste of what it could've felt like, makes them hesitant to let it go, just like Hayama's group.
During Iroha's election, Yukino thought that she would be suitable for student council president such that Iroha would lose with dignity. Everyone knows that if she did, the club will end. So Hachiman forced Iroha to win and hoping with that their "friendship" would still be intact. Though this makes things even more tense.
After that Iroha come again with Christmas request, seeing as how bad things already were, Hachiman didn't want this request to involve the rest of the service club and decided to take matters into his own hands. Even though he did this to save their fraying relationship, both Yui and especially Yukino find it even sadder because they know on why he did it.
When Hachiman realized his methods were wrong, he's tries to request for help from both of them. A bit fed up, Yukino rejected him. After some back and forth between them, Hachiman can't take it anymore and basically admits that he just wants a relationship where they are being honest towards each other and actually become "genuine" friends instead of "playing" friends. Although he worded it in the most vague way possible.
Next description will go a bit further than this genuine monologue, but will provide more context
After they finished Iroha's Christmas request, they sat down together, Yui and Yukino said that Hachiman's request is still not fully completed yet (the part when he said about genuine relationship), but Hachiman didn't get it during this particular time. Yui and Yukino were talking about how they will confess their feeling to Hachiman instead of maintaining status quo. That's because they want to make their friendship genuine now.