Allotments 4 All

A few weeks back, I decided to extend the lawn area, so went through the usual procedure of improving drainage, removing stones and applying a Roundup weedkiller to kill off any weeds. I sowed the grass seed, and much of it has germinated, but a poppy from the border has shed it's seeds all over the new lawn, meaning large patches of the 'lawn' are poppy seedlings! Can I expect this to die over winter? I gave the lawn it's first cut yesterday, so maybe the taller seedlings will die as a result of this beheading. But really want to know if these poppy seedlings will die when the frosts come.

I think that they would be killed by the frost, but in any case poppies are tremendously easy to pull up if you so wish.