Having read through the threads again from searching, it seems that this probably wont be the cause of the handbrake being weak as the spring just takes car of returning the pads to the caliper piston when the lever is let down??
If the spring doesn't return the lever all the way to the stop, then the handbrake/piston mechanism won't take up the slack as the pads wear. Without the wear being taken up, the pads wont press against the disk as much as they should.
Try moving the spring you have to the other side. You may be able to get the caliper to self adjust without the other side wearing too much to need the spring there.
You can easily test the handbrake efficiency yourself if you live in a hilly area.
On your fail sheet, it should say what the efficiency is e.g. 14%. The pass mark is 16%. Sounds terrible doesn't it? A pass with just 16% efficiency! But it is just badly worded. 16% means it has sufficient efficiency to hold the car on a 16% hill. That's all. So if you have a 16% or steeper hill near you, it is easy to test the handbrake before going back for another MOT. It just needs to hold the car on a 16% or steeper hill.
The adjustment procedure for the cable -
Slacken the adjuster nut and lift the rear and apply the lever 4 clicks. You now tighten the adjustment nut until you can no longer turn the wheels manually. Release the lever and the wheels should rotate freely. That'll also show if there's an imbalance from side to side.