Not moving when it's clicked again, why?

The reason it "doesn't move" is because the coordinate that the part should move to is unchanged, meaning that the script is running your code, but since you're repeatedly moving the part to the same coordinate, Vector3.new(5, 0, 0), it looks like nothing is changing, and you can test this by using print statements. If you'd like to see a change on every click, try something along these lines:

local p = Instance.new("Part",workspace)
p.Anchored = true
p.Position = Vector3.new(0,0,0)
p.Size = Vector3.new(4,1,2)

local cd = Instance.new("ClickDetector", p)

function move()
      p.Position = p.Position + Vector3.new(5,0,0)     
end

p.ClickDetector.MouseClick:connect(move)

The script doesn't undergo any major changes, but in this script, it'll move the part 5 studs on the x axis every time you click it, because it is adding 5 studs to the x axis on every click, rather than setting the position of the part to a predetermined coordinate.

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