Is it possible to lose contact with a region on purpose?

Solution 1:

You can loose contact to a region, but not a chosen region as far as I know.

You have the following options:

• Complete a blacksite mission/shadow chamber mission, or blow up an avatar facility
• Complete the first skulljack mission (that is: jack in, kill the thingy)
• Complete the second skulljack mission (that is: jack in, kill the scarrier thingy)
• Fail to respond to the mission to rescue a civilian outpost (or fail the mission by any means). It will make the outpost "regroup", which will sever the contact with this region. Note that this is on a 6 week timer. It might not come around soon enough.
• Use skull mining on a soldier, and successfully get a facility lead. You must have no facility lead in storage. If you already had one in storage, what are you waiting for. Once you obtain one, you need to "research" it, then you attack the facility. The facility lead can also be a reward from hacking a "lamp post", but it seems quite rare.
• Complete a mission that has a contact slot as the reward. Then make contact.

No other missions have a way of reducing the avatar timer as far as I know.