The plug should be a trs connection (tip ring sleeve) and either the female end on the car or the male end on the cable could have a problem keeping it from making the connection.
The sleeve, or the bottom most part of the connector, is typically ground. The ring, or the middle part, is typically the right channel and the tip is typically the left.
If there's a connection issue at the plug itself it would be the center of the connector, the ring. As Davetsbass said it could be the ring is worn and "too skinny" to make the connection. Also, it could be the contact within the female end of the connection is bent inward and does not reach the male plug.