What have i done wrong here with my backpack cloning?

Well, you see, the 'PlayerAdded' (or 'ChildAdded' in your case) event is straight away fired when a player joins the game, meaning the script doesn't wait for the children inside.

What I do recommend is waiting for the Backpack, which can be done so by using the following code:

repeat wait() until newPlayer:FindFirstChild('Backpack') -- I do believe you can just use 'newPlayer.Backpack' aswell

Note, the above code can be written in different ways (eg. using 'WaitForChild' or, a less cleaner way, while not newPlayer.Backpack do)

You can add this wait either before or after your line 4, so it would ultimately look like this:

adminName = "potato12344" -- I was using my name for testing

function onEntered(newPlayer) 
repeat wait() until newPlayer.Backpack
if newPlayer.Name == adminName then 
    local nitro = game.ReplicatedStorage.Nitro -- Nitro is the name of the weapon and IS correctly spelt
    local newnitro = nitro:Clone()
    newnitro.Parent = newPlayer.Backpack 
end 
end 

game.Players.ChildAdded:connect(onEntered)

And there you are. If you still have a problem, just let me know. Good luck.

Please upvote if I helped

PS. I do recommend cloning to StarterGear aswell so that the player has it even after dying