Simple, aerify the crap out of the field, heavily topdress with straight sand and repeat whenever possible.
After the topdressing drag a piece of chain link fence with weight uniformly spread across the leading edge.
Will destroy the cores and distribute the topdressing quite well.
Done it for years.
After about three cycles the turf should respond better each time.
Also, after the aerification and before the topdressing remember to fertilize as well.
I used this process on a minimal budget athletic complex at a local municipality with great results.
The city, in their infinite wisdom, thought they could do it less cheaply and in less than one year royally screwed these fields up.
Now they want me back - but this time they have to pay$$
Nice Try = You Suck Spelled Differently