Hal's never really examined why he couldn't accept Korra as a new recruit. It takes him several tries before he starts getting somewhere.

"Being a Green Lantern is an honour. We're singled out because someone sees something in us and even then some of the new recruits wash out during training, but at the end of it you know that every single one of the rookies who gets through is someone you know will have your back. I don't know about her character, no one picked her, she's a civilian that found a gun and wants to play at a cop."

He'd been struggling with his place in the Corps ever since the incident in the bottle and to see someone just waltz in and claim membership was the greatest insult.