Please stay calm Justme.
You see, this is the problem: in his head, 'I haven't left the children, I've only left you'. That is his split, and it is a delusion that you don't have to do anything about. Time will erode his fantasy that he can have it all his way. So stay dignified, and don't sink to the swamp they wallow in.
In reality, he has left his family. All you can do, is calmly point this out to him.
Don't use the children as a weapon/anything he can beat you (the fat troll who is the cause of ALL his problems and who FORCED him to seek happiness and love elsewhere!!!![violins]) up with blame wise. Please, please, stay calm. They are old enough to deal with this. Make provisions about him seeing his children: it has to be in McDonalds at an agreed time. Because this is what leaving your family means. It means that you have chosen someone else over them, and you have chosen to become a weekend Dad. No house, no own chair, no own coffee cup. Time is on your side if you stay calm! Although you wish him eternal sexual bliss and happiness, this is a consequence of leaving his family.
And he is NOT to upset them by bringing along his shiny new fuckbuddy. If he does, he will be forfeitting their trust and make it far more difficult to see them again.
You CAN stay calm. You CAN stick to the points.
So: Gerald (or whatever his name is):
The children are prepared to meet you in McDonalds on Saturday at 2.30pm. I will make sure they leave the house in good time for that.
You will show consideration for their feelings of being betrayed and left, (note the 'their', not 'my')and you WILL NOT bring along your shiny new fuckbuddy.
If you do break their trust in this way, please consider you will be making seeing them much harder in the future.
Stick to these points. DO NOT get sidetracked. Do NOT attack Lovingopenlegs. Put them on his answerphone preferably so your calmness can't be denied.
If he starts on coming home, reply: you left home and chose another home. This is not your home anymore. (What I said to my H:) this is a place of safety and truth, and you are not to come here and contaminate it. [He was devastated].
He has never met you calm, and steely furious no pushover any more! Time for him to do so.
CHANGE YOUR LOCKS. Do NOT ask your solicitor's advice, permission, or hand over any other power. Do NOT put that question on record. Just do it.