What could cause my sagging neck tendon?

The band can be removed and sutured to the other side if the rest of the face is without significant aging. Otherwise, you would need a facelift.

This is most likely the inner edge of the platysma muscle and it is called a platysmal bank. This is usually addressed by a necklift or full face lift. In either case the inner edges of the right and left muscles are sutured together and this resolves the issue.

Cords in the neck are called the anterior platysmal bands and are the most anterior leading edge of platysmal muscle that tends to shorten, fibrose, and contract with age and facial grimacing. At the time of the face/neck lift the top portion above the voice box is sewn together with a platysmaplasty procedure and the lower portion is actually transected over the cricoid area to separate the fibers so that it does not recur after the lower face/neck lift.  

Platysma bands can be quite bothersome and it sound like you have it all the time. This can be treated with surgery to eliminate the problem.