What's the white wire for on the BT side of the telephone?

[#1]

I've been getting some crackling/strange ringing behaviour on my phoneline.
Rang 151 and BT acknowledged a likely problem within about one minute.
Engineer arrived on site 3 hours later 
Problem caused by builder damaging BT cable externally, about 5 years ago.
Water had corroded the white wire, so that simply touching it caused interference...
New box put on wall.
No charge 
Is it likely that my broadband will improve?
[Just got an automated phone call from BT - "press 1 to confirm fault corrected"  ]
I'm very impressed.

[#2]

BT supply normal phone lines as 2 wires called A and B, so it will be one of those. Details here
As for a faster broadband it depends on what happened to the cable. Does your ADSL equipment sync any faster than it did before?

[#3]

Strangely enough, it didn't show anything different - if anything it has got worse - but it suffered a lot of resyncs during the
replacement of the corroded wire.
I think I'll try for a reboot now. 

[#4]

Also is your SNR/Margin any better?
If not then the fault may only have been affecting the voice frequencies and not affecting the much higher broadband frequencies.

[#5]

The Noise margin has, sadly. increased and the line attenuation is unchanged.
Amazing how a visibly corroded wire can do broadband!

[#6]

The margin may have been set higher at the exchange if it saw disconnections. Hopefully it will recover.
Always best to turn all BB kit off when the wires are being worked on. 

[#7]

Very definitely right HP.