The reason Hawkeye hasn't been part of Avengers: Infinity War's marketing campaign is simple: he's not in the movie... at all. Jeremy Renner has been playing Clint Barton, aka Hawkeye, in the Marvel Cinematic Universe since his brief cameo appearance in Kenneth Branagh's Thor. Then, he became one of the founding members of the Avengers in Joss Whedon's The Avengers. He's only appeared in two other movies since: Whedon's Avengers: Age of Ultron and Anthony and Joe Russo's Captain America: Civil War.
Since Avengers: Infinity War and Avengers 4 filmed back-to-back, there was some overlap, but it was believed that Renner would reprise his role in the third Avengers film. After all, Renner wasn't able to appear in Christopher McQuarrie's Mission: Impossible - Fallout this year due to a scheduling conflict with Avengers. Plus, the Russos themselves confirmed that Hawkeye was going to be on a special journey in the film. And while audiences don't actually get to see that journey for themselves, it's very much implied.
Captain America's team were all sent to the Raft - a special prison for superpowered individuals - at the end of Civil War. While Wanda Maximoff and Sam Wilson escaped with Steve Rogers and went on the run with Natasha Romanoff (who had betrayed Iron Man), Clint Barton and Scott Lang both struck deals with the government to return home to their families. Neither of them are superpowered individuals, after all, which means they rightfully didn't belong on the Raft in the first place.
Clint is under house arrest at his homestead, which was first introduced in Age of Ultron. It's worth pointing out that Clint had retired from the Avengers prior to Civil War to be with his family and his newborn son, Nathaniel Pietro Barton (named after Pietro Maximoff, aka Quicksilver, who saved his life in Avengers 2). Since Captain America called Hawkeye for help in Civil War, that's what eventually led him to be imprisoned, so it makes sense from a character development standpoint why Steve wouldn't risk calling Hawkeye once again, even though this time it was to save the world.
Although Clint Barton wasn't in Avengers: Infinity War, that doesn't mean Hawkeye has left the Marvel Cinematic Universe for good. Given that Thanos was successful in obtaining all six Infinity Stones and wiping half of all life in the universe by snapping his fingers, it's possible that one if not all of Clint's family had disintegrated like half of the Avengers and Guardians of the Galaxy. That would set Hawkeye down the path of becoming Ronin in Avengers 4, which would certainly fall in line with the Russos' long-game plans for the character, but that's something we'll just have to wait and see.
- Avengers: Infinity War / The Avengers 3 (2018) release date: Apr 27, 2018
- The Avengers 4 / Avengers: Endgame (2019) release date: Apr 26, 2019
- Spider-Man: Far From Home (2019) release date: Jul 05, 2019
- Captain Marvel (2019) release date: Mar 08, 2019
- Ant-Man & The Wasp (2018) release date: Jul 06, 2018