Weeds under Leopard tree

We have a large leopard tree in our yard and the grass underneath it is infested with weeds and is spreading across the rest of the lawn. The weeds pop up as soon as it rains and when I spray with bow and arrow the weeds die but leave just mud underneath and then they grow right back 😭 how can I salvage the grass here? It’s often quite damp under the tree as it doesn’t get much sun. This section of the yard really gets me down, I don’t even know where to start. I collect the seed pods and bark and sticks every month or so. Also open to some hardy plants that could be planted undearneath in this area instead of grass. The tree looks really nice and keeps the house cool in summer but I’m starting to think I should consider getting rid of him for morale.