I see you've a young family using the lawn and it depends on what your calling a lawn,
Place for the kids to play is great but the lawn need time to recover if its a lush springy turf your after,
But as a general pick me up for your Lawn try this "it works" start by looking at how hard the dark patches of soil area'a are ie those with little or no grass in the corner area (in your picture) never mind the goal area, and you'll see the ground is dead and this in that area should'nt be, (keep the kids off for a few weeks) and now we start,
cut the lawn first and then using a lawn rake "rake all that dead straw like grass off the area"
The lawn needs Air, so all the area needs tobe entered via a garden fork approx 3 or 4 inch, just put the fork straight into the lawn and pull it out (so your making holes in the turf) do this all over the lawn approx 6 inch spacing and then spread fine sand over the complete area, get a good brass broom and brush the sand over the holes, (the idea is to give the soil an added drainage and break the hard surface up)
water the complete area and leave it a day or two,
Now give the lawn a light (powdered lime) you get this from the garden centre it must be powdered lime not the chunky stuff)
spread the lime over the complete area and and gain water it in, leave the lawn for a day or two and the final step now is to feed the lawn with a lawn feed (verious ones are on the market but they need to be able to get down to the grass roots "hence the work you've just done
I myself swear by EPSON SALTS, ive three dogs "one a bitch" and my 2 acre garden has plenty of lawn both in the shade area's and full sun and the grass is like walking on a carpet,
Now the cutting, from your picture your grass is to short, lift the blade a level "2 to 3 inch" is the better level and cut in different directions every other cut, dont keep going the same way,
Water the Epson salts every 4 weeks and the grass will really look good.
For the size of lawn you've shown in your photo it's not as much work as it sounds.