If you have been the main care of the children until now, and they are as close as you say they are, it’s most unlikely that the court would award him anything other than contact including staying contact.
He applies for residence of the children, you can oppose it on those grounds and also his irresponsible conduct with regard to the insurance. To be honest, if he is driving uninsured, you should tell him that unless he resolves the issue immediately, you will refer it to the police.
If it’s simply an informal arrangement between you whereby he has to child from 4 PM to 8 PM and he keeps the child overnight, you will probably find that the police will not be interested unless there is a court order.
What you would then need to do is apply for the child arrangement order following the process on here
does that answer the question? Can I answer any specific points arising from this?
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