Yard is good, but patch of grass under small oak tree struggling

Do you have a lot of runoff? Maybe try multiple start times for a better soak.

This probably won’t look the best but you might leave it a little longer than the rest and increase watering.

The tree is sucking all the water away from the grass so you have 2 options: either put mulch down as someone else suggested or just get the hose out and put a little water down.

Growing a pretty patch of lawn under an oak tree is a fools errand. You can do it, but the amount of effort (and water) gets ridiculous in my experience. Neatly put some mulch down, perhaps with a landscape border, and put a wrought iron bench underneath as a shade seat.