To which year back in time was Hawkeye sent during the test run for the Time Heist?

There doesn't appear to be any official answer on this. The closest we can do is guesstimate based on Lila's looks between the 2018 version of her at the start of Avengers: Endgame (Image 1) and the version we see in the test run (Image 2).

To me there doesn't appear to be much of an age difference but the test run version does appear to be a little bit younger. However, with not too much of a visible age gap between them at most this was two years (and that's pushing it) prior to the opening scene. This would mean he was sent back to anytime in 2016 - 2018.

She also doesn't seem too fazed that Clint is there and given that he was home during these 2 years on house arrest that makes sense. Before that he would have been constantly coming and going as we see in Avengers: Age of Ultron so that's the small window he was likely there.

It's also worth noting that Lila is played by Ava Russo¹ in every scene she appears in. So whilst they could have made her look younger or slightly older they didn't alter things too much so it's unlikely this was too many years out from 2018.

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_{¹Fun fact she is Joe Russo's daughter: "This is Joe’s daughter, Ava, playing Lila Barton. AND @JeremyRenner’s actual daughter’s name is Ava, so spot on casting there… #AvengersAssemble"}