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D 24
Aerate (hollow core), scarify, top dress and overseed. Probabaly want to wait until September as hot weather forecasts and would need significant watering first few weeks. Someone with more experience may have a different idea
3 options, there was a herbicide available some years ago which would kill the rushes but not grass, sorry can't remember what it was called other than it was an agricultural product, do some research.
Second option use glyphosphate which will definately work as have used it on rushes, yes will kill any grass but not a big job to reseed the dead patches and thirdly dig them out and as second option reseed the patches. I would probably go for this option as if sprayed you will still have unsightly clumps left.
Remembered MCPA and 2 4 D will kill them you just need to source a spray containing these will not kill the grass
Another edit the herbicide is called Enforcer Selective £70 from Pitchcare
Rather than using chemicals they could be regularly strimmed or mown tight into the crown, that kills most things in time. Starving the roots. They look weak already.
Drainage is as you say the main issue, but liming may help as they like acid conditions. It may also help the soil structure a bit.
I'd personally get them to sort the drainage out first as it might keep reoccurring otherwise