I don't think plastic will help much but I always find seed takes better if sown in September. The ground is still cold at the moment but by the end of the summer even the darkest spots are warm. Scratch the areas up and sow the seed during mid september. Cover with a sprinking of earth or compost and keep off it. Water it once a week and it'll grow well into the winter. It'll be protected form anyone walking on it as you all retreat in doors and come this time nest year it'll be bursting into life like the rest of the lawn. More info here