Any time you experience a voltage drop when you turn something on, you are seeing a situation to where the supply isn't allowing enough current to whatever you just turned on. So the voltage has to drop to try and get the amount of current. So with the above ideas, I would most certain trace the wires and have the battery posts cleaned and shiny. And if it still does it, you could have either a defective radio issue or a battery on it's last leg. If the wires are the same from the older radio you removed, does the new radio have the same current demands (fuse size)? If your new radio demands more current then the wires can carry, your wires will also get hot. Be careful if that is the situation. You will need to go to large diameter wires (lower gauge number). JMHO