If the patches are all about that size, just rake out the soggy thatch in those areas - it's a bit too cold yet here in the UK to cut the lawn very short, though you can certainly cut it with the blade/s set higher if its neither waterlogged nor frozen. I'd wait a bit to see if those patches need reseeding - you may find grass grows anyway after a while, but if nothing has arrived by the time temperatures are well above 5 deg C overnight (hopefully by mid/end April) then reseed the patches. You will need to create a tilth about an inch deep prior to sowing the seed; you can do that either by loosening the soil in the patches with a rake, or just dig around the top a bit with a hand fork and tamp it down, then sow seed. Keep watered if the weather turns very warm and dry (though that seems a distant dream at the moment!)
For future reference, there's a rule of thumb in the UK regarding lawn cuttings - if the weather's hot and dry and the lawn was not too long when you cut it, you can let the cuttings fly, but if its wet or going to be wet, collect them up. And particularly collect them once we get past mid August, because that's when heavy overnight dews start. This only applies to cylinder mowers - hover and rotary mowers without a collection box obviously chew the cuttings up and spit them out in wet clumps, and they should not be left behind on the lawn.