Is the spongyness moss or compacted grass?

If moss then you'll need to kill it with moss killer and rake it out and reseed if it leaves bare patches. You'll also need to treat the underlying cause of moss growth.(poor drainage and or poor light)

Drainage can be improved by spiking and sprinkling sharp sand into the holes. If you have trees blocking a lot of the light you could consider thinning them out.

You can also buy lawn seed specially formulated for shady areas.

If it is mainly grass rather than moss you'll need to scarify it. If it's very bad you may need to hire a petrol scarifier or get in a professional to do it for you as the light weight electric scarifiers may not be up to the job . The next two weeks are the ideal time to scarify a lawn as it will have enough time to recover before the winter. Beware it will look a mess before it regrows and if it hasn't been scarified for a long time and there's a lot of dead thatch you'll be staggered how much will come off a square metre( up to one dustbin load!)

I've just been researching scarification and this link is useful

http://www.lawnsciencesouthmanchester.co.uk/cgi-bin/index.cgi?p=lsRC_lawn_heavy_scarification

It probably is moss but a problem I occasionally come across on my gardening round is when I take on a new customer and they've used a mower with a roller for years,never changed the direction of the cut (eg stripes always run the same way) and have never scarified or raked.

The grass tends to grow along the direction of the cut so it grows and lies almost horizontal rather than growing vertically . I can run a mower over it set at say 2 cm but the grass is still at least 7.5cm long horizontally along the ground.The mower is effectively missing most of the length of the grass and you end up with an ever deepening thatch ...the mower grabs at the grass and it looks a real mess

The only solution is to do a very heavy raking/scarification and repeatedly cut until you've got the grass truly down to maybe a centimetre. It looks horrible and the amount of debris that comes off is staggering but if done about now the lawn should look good again in 3-4 weeks .

After that mow against the grain occasionally and across the grain every few weeks to stop it happening again