”First safe country” refers to the EUs Dublin agreement which permitted EU members to send an asylum seeker to the first country of entry in the EU
The Dublin III Regulation enabled the UK to return some asylum seekers to the first country of entry without considering their asylum claims.
Under the regulation, family connections and unaccompanied minors receive a higher criteria rating than 'first safe country'.
The UK's success rate with transfers was low:
Between 2015 - 2018 the UK made 18,953 outgoing requests to transfer people to other member states under the Dublin regulation, from which 1,395 people were transferred - this amounts to around 7% of outgoing requests by the UK resulting in a transfer.
During the same time period, 7,365 incoming requests were made to transfer people into the UK under the Dublin regulation, from which 2,365 people were transferred - this means that the UK accepted around 33% of requests.
EU regulations also allowed for asylum seekers to be distributed across the EU instead of concentrating on the external facing EU countries
Angela Merkel famously accepted over a million asylum seekers 2015/2016. The EU's new solidarity pact comes into force fully this year.
The Brexit negotiations could have included agreements to remain in schemes such as the Dublin agreement (and Europol networks etc) but it was struck off
Remaining with a view to what? Reducing the number of asylum seekers we process overall or numbers similar to what we have now? By way of comparison, France received over 150,000 applications last year.