I do a fair bit of karting and must say that Sheffield isn't particularly well served for it.
There's Parkwood up near the Ski slope, but it's too expensive and last time I went there they were stopping 'arrive and drive' sessions, as well as regular group bookings.
There's an indoor place off the Parkway on the Attercliffe side. It's near Macro and the old Yorkshire Electricity building. Looks OK but never driven there.
Further afield things are a bit more fun. The best local-ish place (IMO) is Amen Corner near Edwinstowe (out towards Worksop). They have a good size proper asphalt outdoor track with overtaking opportunities and a couple of ars*-clenching corners. They also got well maintained twin-engined karts that get up to a fair old speed. And it's not expensive either.
There are some good places in Leicestershire too, but that's probably a bit far.
Avago in Rotherham isn't very good. I've only been once, but it's possible to drive a full lap without lifting or braking, which is lame