While the joke has been made many times, even by the character himself, that it doesn't make sense for Clint Barton aka Hawkeye to be on the Avengers considering he's the only one without any powers, he has still managed to consistently prove why he deserves to be on a team with every Marvel Cinematic Universe movie he's appeared in.
In addition to being an expert fighter and archer, Barton also has a variety of trick arrows that allows him to compete on the same level as the other members of the team. While fans could see even more trick arrows in the Hawkeye series when he passes the bow to Kate Bishop, the ones they've already seen have been pretty cool.
Explosive
Out of all the arrows in his quiver, the most popular one Hawkeye uses besides the traditional ones, is his set of explosive arrows. The arrows have explosive tips that can be detonated using a button on his bow and have been used in almost all of his movie appearances. They were strong enough to do serious damage to the Helicarrier, even destroying one of its engines, as well as blast Loki from the Chitauri chariot he was riding in during the battle of New York in The Avengers. However, some characters have been able to resist them like in Captain America: Civil War when Black Panther was able to catch two that were fired by his head and remained unfazed as they went off, thanks to his vibranium suit which protected him from the blast.
Trap
Trap arrows are a set of arrows that can be placed on opposite sides of each other in advance and can deliver an electric shock to anyone that passes through them. Clint used the arrows to temporarily incapacitate Vision when he came to rescue Wanda from the Avengers compound where she was being held under Tony's orders in Captain America: Civil War. Despite Vision's vibrainium body, they were able to hold him back for a few seconds before he used the mind stone to destroy one of the arrows, disconnecting the charge and freeing himself.
Hacker
While most of Hawkeye's arrows are solely used for combat situations, he does have a few that serve purposes beyond being used in fights. One of those being his hacker arrow which is an arrow that contains a solid-state computer drive in the tip and can upload a specific code onto any device it's plugged into.
While under the mind control of Loki, Hawkeye was able to use it to shoot the arrow into the main command center of the Helicarrier and shut down another one of the ship's rotors which had to be fixed manually.
Burst Shot
Hawkeye's quiver can only hold so many arrows and there are often more enemies than he has arrows for but luckily he has some that help with that as well. Hawkeye has a set of burst shot arrows that have secondary projectiles that are stored in the arrowhead and get ejected after hitting its intended target. The arrow works best when Clint is outnumbered and going up against a group rather than an individual which is why he used it against the Chitauri during the battle of New York, allowing him to take out several of the aliens using the least amount of arrows possible.
Thermal
Sometimes the enemies Hawkeye fights will have armor or plating that would be too difficult for a normal arrow to pass through which is why he also has thermal arrows that he can use when the object he needs to pierce is too thick. The tip of the thermal arrow has a chemical mixture that, when activated, can melt through almost any material including most metals. It was one of the many arrows he used during the Battle of New York, aiming it at a Chitauri chariot, causing it to overheat and crash into a nearby building.
Grapple
Since he has no powers, getting around from place to place is much harder for Clint since he can't fly like Iron Man and Thor or make giant leaps like the Hulk. It also makes it more difficult for him when he needs to get out of dangerous situations like when he had to jump off of the building where he was perched for most of the battle in The Avengers in order to escape the horde of Chitauri chariots that were heading towards and firing at him. Luckily as he was falling, he was able to fire a grappling hook arrow at the side of the building, allowing him to swing through one of the windows and land inside, protecting him from the attackers.
Pulse
While Hawkeye usually doesn't miss, he does have arrows that make it easier for him to hit his target or even multiple targets at once. Hawkeye's pulse arrow emits a 360-degree shockwave that is capable of sending those in the general vicinity flying backward or at the very least, knocking them off their feet.
Instead of needing to be precise, he can just aim in the general direction of the person or people he's trying to take out and still manage to deliver a non-lethal blow to them. He used it in Avengers: Age of Ultron to take out the remaining members of Klaue's crew while the other Avengers were fighting Ultron and the Maximoff twins.
Electrical
Hawkeye's electrical arrows, which emits a electric charge on whatever it hits, proved useful in Avengers: Age of Ultron when he mostly used them to destroy multiple Ultron drones while fighting the android's army in Sokovia. He also used a less powerful but highly effective version of the same arrow on Wanda Maximoff earlier in the movie to prevent her from using her mind-warping abilities on him, as she had on the others, by attaching it to her forehead. The electrical arrows briefly made a return during Captain America: Civil War as well when he fired one at Vision to prevent him from attacking Falcon during the airport battle.
Flare
Much like the hacker arrow, Hawkeye's flare arrow serves a functional purpose rather than a combat one. The flare arrow emits an extremely bright light that can be used as a signal or to bring light to dark areas which is what it was used for when it first appeared in Avengers: Endgame. After finding himself trapped in the tunnels beneath the Avengers compound after it was destroyed by Thanos, he fired a flare arrow through it, revealing that a group of Thanos' Outriders had made their way down there to retrieve the Infinity Gauntlet.
Scatter
The scatter arrow is a singular arrow that can split apart into 6 smaller fragmenting arrows so that it can hit an opponent multiple times after being fired. The arrow first appeared in Captain America: Civil War and was the one Ant-Man rode on, during the recreation of the classic comic book moment, before it was fired at Iron Man so that he could secretly land on him, sneak in, and shut down some of his armor. Even though Tony was able to blast the arrows out of the sky before they could hit him, they served their real purpose of being a distraction that Scott could use to his advantage.