That's actually a pretty good-sized patio! And with landscaping and a tree! spiff.

Anyway, I'd leave it as is for now -- put your table and chairs at the end near the house/grille and let the kids run around in the less structured space around the tree. The path is enough to draw one on, but I wouldn't want to seal up the rest of the space. (I guess I would repair it with the loose bricks, though, if there's an actual gate back there.)

Find some shade-tolerant hardy plantings -- we have a lot of myrtle around a tree in a patio like this, and it looks nice in all seasons, has flowers in early spring, and can take some stomping. There are other ground coverings too that can take a bit of walking on like grass (but with a chance to make it, heh). You might or might not be able to get anything more substantial/interesting to survive near the edges away from the play zone. Depends on the foliage density of your tree.

Great space!

A full circuit of the yard for a tricycle or roller skates could be made with paver stones. Being able to go in circles is a lot more fun than riding full tilt at a wall and having to back up. Then divide remaining area into soft area for kids with grass or dirt and a hardscape to keep the grill and chairs on. Add an umbrella for shade. A foldable card table for occasional meals outdoors would be easy to use and store, or try a real picnic table the kids can also use for painting, clay, and other messy outdoor activities. Staple a tough vinyl tablecloth on it for easy cleanup and quick replacement.

I would deck the whole thing and use big pots for planting , that way the water runs off and still waters the tree and the whole space is usable for entertaining and for the kids to play. IMO the tree really takes so much space I think i would remove it and get one that provides shade but the trunk is straight and the foliage starts a about 6.5 feet you could then even put a bench seat in the back corner.

Don't touch that tree! It will provide summer shade and bring wildlife into your garden. If you want to expand the patio, widen it in the area where the toys are in the picture. Vines on the fences would add color and texture without taking up much space. A couple of large glazed pots filled with colorful plants will add pizzaz. How about a couple of bird feeders? You might invest in a smallish round table with an umbrella and some moveable chairs for summertime lounging. Think color. Think compact. Think comfort. Summer is coming!

build a tree house!

I wouldn't worry about paving your patio so kids can pedal in circles -- that's what the sidewalks out front (on the way to the playground!) are for. This is an urban setting, after all; back yards are for yard space, not blacktop.