Shady plants.

The dryness will be the problem - most shade loving plants like dampish soil, unfortunately, plus the willow roots will be extensive in that area. There's always Ivy, but I wouldn't recommend it! How about cutting a circle (if its in the lawn) all the way round and just using fine bark chips to tidy it up?

I under planted my weeping willow with geraniums as they seem to grow anywhere.
I have just chopped them down now so they can have a second flush.

The willow also gets a trim to stop the branches going all the way down to the ground.

It also gets a chop at the top at the end of the year to keep it a managable size.

Ajuga could work well. I've tried to get rid of some this summer by really starving it of water to try to make it die, as it was where I didn't want it. But the plan failed and it is still there, thriving and well.

Ajuga is great I potted up loads this year to give to friends.

Many of the hardy geraniums would be fine there.

try somehing like Mitella, its under my beech tree and is a lovely vibrant green. i have the aforementioned plants as well. dont forget plants like cyclamen and colchicums for autumn flowers. the spring is a world of bulbs etc if you want to.

Hi Seaburngirl, Is Colchicum happy in shade, even full shade?

well mine are under the beech tree canopy and the leaves grow before the canopy opens so i guess yes they are happy.

Beech is naturally late into leaf, so presumably the leaves have a chance to make their food in time.

I suspect that could be the reason. they have been there 8yrs now.