Dear friend
That 4 prong connector attaches to that rectangular silver thing which is a RELAY.
It is needed a photo of the connector opened to look the position of the cables, and in that way have an idea for what service or function it is used that relay in your electric arrangement.
In principle I do not think is related to the ignition, but you never know.
There are full diagrams of the electrical wiring of your car on the technical section of this forum. They are very usefull. I think you have to become a member to access them, which cost only 30$ annually. Believe me it is worth the investment.
A relay has 4 terminals in principle. The control terminals are numbered 85 and 86. The brown cable in your connector should be on the pin 85, (- signal). On the port 86 you need a positive activation signal.
Then ports 30 and 87 are what is called the switched circuit. Port 30 is the source signal which has usually a thick cable, and port 87 is the output switched signal. Port 87 has current only when control terminals 85, and 86 are excited at the same time. I mean port 85 is connected to ground and port 86 to a positive trigger source.
Have a great day.
Best regards
Eng. Leonardo Peterssen
Www.wiredoktor.com