The closest is in Los Alcazares/San Javier, near the Airport.
Click the picture below to go to their website.
There is a cafe-restaurant there too. To get there head for the Airport but at the last roundabout before the airport turn right and its a little further down there, next to Discoteca Mana. Prices start from 6 euros and they different karts for different ages.
There is also a Go Karting track on the way to Cabo de Palos /La Manga Golf resort. They have a again a variety of different karts including dual Karts for smaller kids and an adult, or dual karts for 2 adults! Click pictures below to go to website and see more pictures.
Prices start from 8 euros and there is a cafeteria on site. Open 10.30 am till 12.30am.
Follow the signs to la Manga ( MU-312) and just before Cabo de Palos is exit 13 . The Go Karting is right next to the MU312.