Thinking quick, Hawkeye pulled out a sonic arrow tip that he kept in one of his pouches. To block out the mind-control, he bit down on it and turned it on. Deafened from the stunt, he was able to subdue Mockingbird, escape, and take down Crossfire. Afterwards, he politely nodded his head when Mockingbird started hitting on him, but she eventually caught on to what was wrong and shortly after, the two were married.
Clint had lost 80% of his hearing, so he started wearing a hearing aid and that was his status quo for a good 15 years or so.
The big Onslaught event in the comics led to a plot device where the Avengers were sucked into oblivion. Franklin Richards used his reality-rewriting powers to bring the heroes back from the brink, but it was used as something of an in-story soft reboot.
At the time, Iron Man was a teenager and Wasp was a literal insect woman, so when Franklin brought the Avengers back, he brought them back the way he remembered them. Since he wasn’t aware of Hawkeye’s hearing problems, Clint was recreated with healthy eardrums.
In the acclaimed Matt Fraction and David Aja Hawkeye series, which the MCU show is drawing a lot of inspiration from, Clint was jumped by the assassin known as Clown (who’s also set to show up in the Disney+ series). Clown stabbed Clint in each ear with his arrows, then shot Clint’s brother Barney. While neither were killed, Barney was resigned to a wheelchair and Clint was rendered deaf.
A flashback showed that Clint had been dealing with hearing issues since he was a boy, a side-effect of his abusive father’s handiwork. That made him reluctant to use sign language to communicate with Barney, as it reminded him of worse times and stung his pride.