How can I finish this ? I can't figure out how I can make the push with Bodymovers

Let's look at what you're asking here.

You say you want the target to be hit when:

1. The mouse is over it

2. X is pressed

Let's check for these two conditions. It's easier to start the check from a key press using ContextActionService.

local ContextActionService = game:GetService("ContextActionService")

local function pressedXHandler() -- feel free to change the function name to anything you want
    -- this function will fire when X is pressed
end

ContextActionService:BindAction("nameOfAction", pressedXHandler, false, Enum.KeyCode.X)
-- change "nameOfAction" to whatever action name you want

Now we have bound the X key to pressedXHandler. Now, we should check inside the handler if the mouse is hovering over the target. We can use the :GetMouse() method, like you already have, to get the mouse of the player.

I'm assuming the 'target' you are looking for is other players - so I will make the target look for a character object. If that's not the case, feel free to adjust the script and change the target.

local ContextActionService = game:GetService("ContextActionService")

-- need to reference the mouse
local player = game.Players.LocalPlayer
local mouse = player:GetMouse()

local function pressedXHandler()
    local hitCharacter = mouse.Target.Parent
    local hitHumanoid = hitCharacter:FindFirstChildWhichIsA("Humanoid")
    if hitHumanoid then
        -- The mouse was hovering over another player, perform action
        -- Assuming you want the hit character to take 25 damage:
        hitHumanoid:TakeDamage(25)
    end
end

ContextActionService:BindAction("nameOfAction", pressedXHandler, false, Enum.KeyCode.X)

This script should be sufficient. I hope it helps :)

EDIT: Adding Cooldowns (Debounce)

Add a variable in your script that only allows the pressedXHandler function to run when the variable is set to true or false (usually depending on the name of the variable)

Here's what I mean:

local ContextActionService = game:GetService("ContextActionService")

local player = game.Players.LocalPlayer
local mouse = player:GetMouse()

-- Add a debounce variable
local debounce = false

local function pressedXHandler()
    if debounce then return end -- This will stop the function from running if debounce is true
    debounce = true -- Set debounce to true so the function can't be called again
    local hitCharacter = mouse.Target.Parent
    local hitHumanoid = hitCharacter:FindFirstChildWhichIsA("Humanoid")
    if hitHumanoid then
        -- The mouse was hovering over another player, perform action
        -- Assuming you want the hit character to take 25 damage:
        hitHumanoid:TakeDamage(25)
    end
    wait(1) -- This is the cooldown time
    debounce = false -- Make sure to set debounce back to false in the end
end

ContextActionService:BindAction("nameOfAction", pressedXHandler, false, Enum.KeyCode.X)