Hi Sara - Roundhay Parkrun host a specific junior event EVERY Sunday at 9am. The route is 2km and I think all on grass. This might be a bit long, though. See link.
www.parkrun.org.uk/roundhay-juniors/
Pudsey carnival on Sat 17th May will host a few sprint races, courtesy of a fantastic local running club... You might want to search for other galas and carnivals closer to TN?
Also, as part of the 10k event, the Club will host a kids race around Pudsey Park scheduled for 22nd June. Approx 400m. Bit later than spring but might be an option?
NtS
(Neil the Scot)