Pretty sure the picture is not the hairspring.
That is the mainspring. Mainspring stores energy and while it might affect timing if it doesn't release the stored torque correctly, but generally the problem is not there.
Timing is controlled by escapement, hairspring and all the bearings ( jewels ) in that area.
If it is consistently 10minutes off, probably it only needs an adjustment. If not, i would look at the hairspring and the jewels in the escapement area. If it stops at 10:30 PM, I would look at the date mechanism too. Got to also make sure the hour/min hands are not touching the crystal/dial or each other.
Waiting for update here.