A further four boats carrying 65 migrants have been picked up in the English Channel, the Home Office said.
Border Force patrols found the boats travelling towards the UK on Sunday, a day after at least 151 migrants on 15 boats arrived on the Kent coast.
On Saturday the Home Office said it had asked defence chiefs for help.
Immigration minister Chris Philp said he would be in Paris next week to demand stronger measures from French authorities.
He said he wanted to make the route "completely unviable" so migrants "will have no incentive to come to northern France or attempt the crossing in the first place".
He said he also wanted to "return as many migrants who have arrived as possible", adding there were "returns flights planned in the coming days".
More than 500 people have been intercepted crossing the English Channel in recent days, including 235 - the record for a single day - on Thursday.
The Home Office has said the Royal Navy could be brought in and there has been talk of copying Australia's controversial policy of physically pushing back migrant boats.
'Reckless and dangerous'
Ex-Labour home secretary Jack Straw said on Saturday any attempt to use those "push-back" tactics would not work and could lead to boats capsizing.
Human rights organisations, including Detention Action and Amnesty International UK, condemned the idea of boats being forced back into French waters.
Amnesty said deploying the navy to the English Channel to prevent people crossing to seek asylum would be "unlawful, reckless and dangerous".
The 23 youths were taken into the care of Kent County Council, on top of the 70 who arrived in July.
Those figures do not include those travelling with their families. The Home Office has refused to confirm the number of children arriving.
Since January 2019 at least 5,800 people have entered the UK on small boats, and about 155 have been returned to Europe.
The Home Office blamed current regulations - which determine where an asylum-seeker's claim is heard - for the comparatively low number of people to have been returned to Europe.
Mr Philp added: "We will also continue to go after the heinous criminals and organised crime networks putting people's lives at risk."