But if it says "VWC + Connections" I'd say (I may be wrong) you could hop from Lime Street to Central on it.
As Tony Mac said, you'll need a new ticket from Liverpool Stns to Kirkdale (£1.90 sgl)
for the dates I looked at, advance tickets to local stations are now £4 - £9 more expensive than to Lime St.
That was definitely not the case when I last booked one.