He wasn't calling her Hawkeye. That was all there was to it. She could keep the name with other people, but as far as him referring to her, she was lucky it was just squirt. He reached into his quiver and tossed her about half his arrows. That should get her by for now.
And Mister Doof? What was this? Clint shook his head at the silly girl, but regardless kept his smile. "Alright. Just try and stay out of trouble in the mean time, ok? And don't get shot. That's the last thing we need." He teased as he jumped back up the fire escape and climbed away making his exit. He couldn't ask her not to do her thing.
"Remember. Next week. Tower. And don't be surprised if you get a phone call first." He finished at the top of the building. He turned then and made his way back across on patrol. Hopefully she was smart enough to not get herself killed.