Vauxhall's place is a go-kart track
Wirral only has Pathan Park now afaik, Sundays £20/day for your own Quad https://www.pathanpark.com
There are others by Chester and Frodsham
There's a scramble track not far from Cheshire Oaks on Shellway Road whose status I'm not sure off, either its legal or "officially" overlooked. They may allow quads, I might have seen some there, not sure. I presume its some sort of club as there are parents there and everything, it used to be run by Base Motopark until council sold it off in 2014.