Hello David, I hope that when you say manure you mean composted manure, dark and reasonably crumbly, often 12-18 months old. Otherwise, if bedding is less rotted that encourages slugs, whether covered or not.
You need membrane only if beds are currently weedy and you want to kill some by light exclusion. Otherwise few should germinate over winter. Its up to you, I leave uncovered here. Also from now until end Sept you can sow mustard Synapsis alba as green manure which is killed by frost, so no need to mulch or.. dig in!