Would allowing asylum seekers to work legally, be another pull to come here?
The response to that is to process asylum applications within (say) 4 weeks. Once accepted you can work, rejected it's back to (presumably) France.
However in a country where it can take 6 weeks to see a doctor and 1 year to see a consultant, that level of resource deployment brings it's own problems.
And tackling the problem in the UK is a bit dim anyway. If we really wanted to sort the problem (spoiler alert: we don't) then working to improve the lives back in the countries people are fleeing would be a much better investment. However it would be much harder for people to milk the system for their private companies that have infected the process that way.
It would be entirely possible to create a graph showing that the number of people seeking asylum is indirectly proportional to the UKs foreign aid budget.