Zarch writes

I imagine the
trees roots are taking what little moisture there is out what little
soil there is and leaving the grass with nothing??

Got it in one!

I could i try to raise the level of the soil in this area??

I think the tree would probably appreciate this ;-)

It's probably a matter of removing either the tree or the lawn. How
would you feel about doing something else under the tree? Put an area of
cobbles with a seat on it, grow cyclamen (which flourish in dry shade),
grow some ornamental ivies (let them grow along the ground but not up
the tree)?
--
Kay