You don't say how big the deck is as a portion of the yard. Also would help to know how high up your deck is and do you have space underneath. Is the yard just grass or lawn?

From what I understand of this, the problem is the soil underneath - and probably your whole yard - has poor aeration when it gets wet and whatever organic stuff is in the soil goes anaerobic (that putrid smell is stuff rotting without air).

So: whatever you do needs to help get more air in the soil, reduce the waterlogging of the soil by either helping water drain out / through the soil better or reducing the amount of water that gets in the soil.

This might mean some combination of making sure drainpipes go further away from the house, or a french drain/dry well to improve the drainage.

An 'organic' version of this might work - deep-rooted plants to break up the soil, extra compost or wood chips or something to work its way into the soil, and would be good for the soil anyway. If you can till and amend the soil in the rest of the yard, that should help some. Might take a while to work though and not be enough without some improved drainage.

I don't know how well the improved drainage on its own will work if the soil is really dense and compacted. Under the deck, it's possible a thick layer of wood chips would buffer some of the water going into the ground below. May not be needed though, as it sounds like the smell is from the entire yard.

Good luck, just some thoughts.

@armoured sorry, I should have included more details: I'm not sure of the exact size of the deck but it probably makes up about 50% of our back yard. The clay underneath is on a downward slope from the house) but has about a 50cm/20inch high opening/access at the far end of the deck which means there is technically a fairly tight crawl access.

The rest of our backyard (grass) has no smell at all, it just seems to be the clay under our deck. There are no plants or grass under the deck, which would add to the lack of aeration under there too.

The deck isn't high enough to plant anything under it I don't think.

We have tilled up the ground around the side of our deck (in preparation for some landscaping) at the upper end of the ground slope but it hasn't had any impact on the drainage/smell from the clay underneath the deck at all.

Limited options without taking part of the deck apart. Divert as much water in rainy seasons away as you can, and I guess a possibility that a french drain/dry wall outside the perimeter of the deck might help, but it may depend on the soil there. If you've recently tilled or dug around the side of the deck, that part might not be that hard. If you have a landscaper, they might be able to help.

But a tricky problem to solve with the deck already there.