In the aftermath of Thanos having snapped away half the universe, including Hawkeye's entire family, Clint Barton decided that the best course of action was to let loose his pent-up anguish and frustration against criminals in brutal fashion. Barton took on the new identity of Ronin, who wielded a sword instead of a bow.
Hawkeye has recently confirmed the inclusion of Maya Lopez, a.k.a. Echo, who in the comics was the original Ronin. It's very possible that the MCU could directly adapt this storyline, explaining that Echo comes to New York to confront Barton about using her alias to kill criminals. In a new set photo from Hawkeye, Clint still has a constant reminder of his time as Ronin through his arm tattoo last seen in Avengers: Endgame.
Jeremy Renner has posted a set photo from Hawkeye, presumably from a makeup station, showing him with the same tattoo Clint Barton adorned in Avengers: Endgame.
Below is a comparison of the same tattoo last seen in Avengers: Endgame, showing a few distinct differences between the body art designs:
The most obvious change is that there is a much wider variety in colors used in the Hawkeye tattoo, while Endgame primarily used black ink with some green. Overall, the tattoo in Hawkeye seems far more detailed, too, at least in terms of finer details and shading that's applied to the shogun helmet and skull wearing it.
Finally, while it could be due to how Jeremy Renner is leaning on his tattooed shoulder in Endgame, it also seems like the tattoo in Hawkeye was moved down his arm slightly.
WHAT THIS MEANS
No, this certainly doesn't mean that this tattoo in Hawkeye is something Clint Barton had changed in-universe, just that it seems like Marvel Studios themselves have decided to redesign it for several plausible reasons. The most likely is that, in Avengers: Endgame, Hawkeye's tattoo was never a focus in any scene or had need to be as detailed as the one fans will see in Hawkeye, which will likely involve closeup shots of it. Makeup artists may have also been allowed to add more color to the tattoo to make it more visually interesting for fans, now that it'll likely get more focus.
Another explanation behind the design change is that this specific tattoo Jeremy Renner has is for a flashback sequence in Hawkeye when he first got it. The colors could have faded with time over the five-year time-skip before Black Widow recruited him for the time heist. However, if that were indeed the case, then the black ink should also be fairly faded in Endgame, not just the colors.
So, it isn't a certainty whether this tattoo is meant to be one for the present or the past, but it would certainly be interesting what inspired Clint Barton to get such a tattoo and start his path down becoming a much more violent Ronin.