Trying to check for child and only execute line of code if child isn't within parent. How?

From line 20 of your first script:

local removed = workspace.Cannon:FindFirstChild("Power",true)

if removed == true then
    wait(2)
else
    TemplateL.Opener.Touched:Connect(Open)
end

At line 22, I can see that you're trying to compare a function that returns an Instance (if found) to a bool - which means the local variable "removed" will never be true and therefore connects the function to a Touched event to the Opener without care.

--From line 22
if removed then --does the children exist? Yes? Yields for 2 seconds.
    wait(2)
else --if the child doesn't exist or nil?
    --do your thing
end

--Instead of comparing local "remove" to a bool value (which is like comparing a variable to an exact value), you can instead do this method to check if the item exist or not a nil/false value.

You can also use a logical operator that determines if the child does not exist (also returns the opposite value if relevant):

--From line 22
if not removed then --does the children NOT exist?
    --do your thing
end
--Only use if you don't want to wait for 2 seconds if the child exists.

Not yet.

Now, those should've been a solution to your question. But not yet.

Well... I'm not gonna lie, but the answer above is just a minor fix of how you tried to compare a function that returns an Instance to a bool value, which is of no difference to comparing chalk and cheese. Now here's the real way to solve your problem:

Check if Power is a child of workspace.Cannon right when the local Open() function is called.

Short demonstration:

local function Open()
    local power = workspace.Cannon:FindFirstChild("Power", true)
    if not power then --"Power" is not a child of workspace.Cannon
        --do your thing
    end
end

--Or, you can use "return" to stop the following execution of the function during the call.
local function Open()
    local power = workspace.Cannon:FindFirstChild("Power", true)
    if power then --"Power" is a child of workspace.Cannon
        return --ignores the rest of the function
    end

    --do your thing
end

Since your script only run once (except the connected function which will be fired by other sources), your attempt of finding the child will only perform once. What if Power is not the child of workspace.Cannon during the execution? It will wait for 2 seconds, then run the following lines of code without looking back. Pretty inconvenient, isn't it? Toggling the script on and off would be quite annoying.

To make it work, connect the function to the Touched event immediately after the function is made instead of giving certain conditions that are only checked once.

Here's a quick fix.

local Activat = script.Parent
local TemplateL = Activat.TemplateL
local TemplateR = Activat.TemplateR
local ActivCode = game.ServerStorage.Fireballclone
local ActivCodee = game.ServerStorage.Power

local function Open (part)
    local removed = workspace.Cannon:FindFirstChild("Power",true) --moved here
    --check if the "Power" is avalable, it will ignore unless it is removed.
    if removed then
        return --ignores the following lines of code in the function, returns nothing
    end

    local parent = part.parent
    if game.Players:GetPlayerFromCharacter(parent)then
        local activater = ActivCode:Clone()
        local activates = ActivCodee:Clone()
        activater.Parent = game.Workspace.Cannon
        activates.Parent = game.Workspace.Cannon
        --cannons are created.
    end
end

--This thread only executes once so... after this its job is done. In case if this one loops then it is necessary to remove all connections your created before starting a new loop sequence using :Disconnect().
--For now this is not necessary, moving on

--Connects the Touched event, without any conditions, because it is already accounted by the function above.
TemplateL.Opener.Touched:Connect(Open) --creates a connection

--The rest of your script goes here
--This is your script, so you have the rights to edit it if there are parts that you don't prefer doing.

You can do the same to the Close() function you made.

Sometimes my words can be really confusing (with the fact that I'm not a native English speaker). Nonetheless, I hope that'll help you, and if that, umm, doesn't, then sorry in advance!