According to a Guardian report of where the Cummings were seen, it wasn't Barnard Castle at all; it was Startforth. Now, admittedly crossing between them takes about 20 seconds on foot over the 16th C bridge, but Barney is in County Durham and Startforth was in North Yorkshire, for example, until transferred to County Durham administrative and ceremonial purposes on 1 April 1974. As with all these places, they don't hold with this new fangled stuff, and consider themselves from Yorkshire. Mixing them up, for the locals, is about as risky as mixing up Newcastle and Gateshead.

Anyway, Startforth is south of the river, and GMK is north of the river, and would be a substantial walk for the child. Depending on how they got to Startforth and where they parked, Dominic Cummings might have gone, but I doubt if the child would. It would make more sense to park north of the river in Barney if visiting GMK was part of their reason for being there.

So my inclination is to suspect they did not visit GMK on this occasion, though of course that is largely conjecture.