Grass under hens is always going to be eaten and scratched down. Can you rotate the areas they use? For the weeds, post a pic on here and we can try to identify them, which will help with eradication advice. If it is only a small area, then digging it over and weeding it as you go could be the best way. You could use a rotavator (you can hire one by the day) but you would need to weed it first. As well as getting rid of the weeds, this would help drainage if you dig deeply. Once cleared and dug/rotavated then get your grass mix sown. Depending on where you live, you might be OK to sow grass seed now, but here we would wait until March when all grass is growing to plant fresh seed. Let the grass get properly established before letting hens back on, then remove them before the grass is killed off again so it has time to establish properly. Chicken droppings are very strong so will burn the grass, as well as the scratching bringing the roots up.