How to cheek that a the part clicked have the name Cartridges?

The Mouse.TargetFilter part was useless you can just do Mouse.Target.Name so your code should now be this.

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

function clickObj()
    if mouse.Target ~= nil and mouse.Target.Name == "Cartridge" then
        mtarget = mouse.Target.Name
        print(mtarget)
        down = true
    end
end
mouse.Button1Down:connect(clickObj)

function mouseMove()
    if down and mtarget then
        local posX,posY,posZ = mouse.hit.X, mouse.hit.Y, mouse.hit.Z
        mtarget.Position = Vector3.new(posX,posY,posZ)

    end
end

mouse.Move:connect(mouseMove)

function mouseDown()
    down = false
	mtarget = nil
end
mouse.Button1Up:connect(mouseDown)