Welcome to the forums and congrats on your new allotment, even if it is a bit challenging ... !
Glyphosate works by being absorbed through the leaves of the plant and taken down to the roots. This means that you need enough leaves to start with, and the plant needs to be in active growth, and so you shouldn't cover the grass as soon as you've applied the weedkiller. It should be one or the other.
You might want to use a cover after a few weeks, once the grass has gone yellowed or brown, to minimise growth from roots that weren't attached to grass that was killed.
If you do apply weedkiller, then don't do so if the grass has dew or water on it, or if it's due to rain in the next 6 hours, or if it's breezy / windy.
Good luck!