They can't.
To answer the titular question, the Charlotte Hornets cannot do anything to make up for losing the NBA Draft Lottery. Cooper Flagg is that special of a prospect that there's really no making up for it. Dylan Harper is the next-best thing, but they're probably not getting him anyway.
But they can do things to ensure this offseason isn't a loss and make sure the Hornets aren't defined by not having Flagg on the roster. There is a path to improvement here, and while it may not be as huge of an improvement as drafting Flagg would be, it will help a lot.
First things first, they have to nail the draft. They're likely going to be picking the remaining player between Ace Bailey, VJ Ecgecombe, and Dylan Harper. Two of them will go second and third overall, leaving one for the Hornets at number four. Kon Knueppel and Tre Johnson are also in the mix.
I've talked myself out of Bailey as an ideal pick for them here. If he falls, it won't be the end of the world because he is talented; he just doesn't fit the roster as well as others. Edgecombe or Knueppel (bonus points if they can trade back a hair for Knueppel and get some picks) would be my choice at fourth overall to really nail it.
They also have two second-round picks early on, numbers 33 and 34. They can use them and get some decent prospects, but I'd really consider packaging them to move back up into the first round. It's a deep draft class, and someone will fall into the 20s that the Hornets would love to get.
Maybe Carter Bryant or Asa Newell slide into the range that Charlotte can move back up into. If not, they can also target Nique Clifford or Walter Clayton Jr. at that range. If they opt to use the picks, then Drake Powell, Adou Thiero, Alex Condon, Johni Broome, or Maxime Reynaud would be quality pickups.
Doing the right thing with the picks is how they win the offseason. I'm of the mind that trading up into the back sector of the first round is the smarter move here, but I'm totally fine with taking any combination of two of those prospects mentioned.
That's only one part of the puzzle, and unfortunately, the other big piece is restricted. The Hornets have virtually no cap space, so they're not going to sign anyone of note for the most part. Expect the rotation to remain largely the same.
They should, however, make some trades. Josh Green should be the first to go. He's not working out. He's not terribly expensive, but that would free up some cap space and get some draft capital, although a second-rounder or two is all the Hornets can reasonably hope for.
Second, Miles Bridges should go. This will free up a lot of cap space, although the Hornets may end up taking an expiring contract on to make it work. Still, they'd get some flexibility and some draft capital back.
If they really want to get crazy and restart the rebuild, they can trade Mark Williams. I've also talked myself back out of trading him since it failed once. The Hornets are better with him, and he's still young. He can be a part of the core, but if they traded him for another decent haul, it would undoubtedly make this a strong offseason.
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