he is lying about me to them already .
Even if it gets written into the Order you can't stop sneaky behaviour. You just have to allow your children to see and work out themselves that he doesn't always tell the truth.
And if granted, how does communication work regarding child access, does this have to go through court if mediation isn't advised?
Your solicitor should be trying to negotiate with your DC father, directly if he's not represented, or DC father's solicitor. Then if there is agreement get that written up as an Order.
If there is no agreement then you have to go to Court leaving it to your barrister to try to negotiate an agreement with him, directly if not represented, or his barrister to turn into an Order on the day. And failing that the judge will rule.
It depends on the nature of the abuse on how contact is arranged. So I know people who even though there was no abuse that went to Court but had allegations been made about them, so had restrictions written into their Child Arrangements Order.
So parties can be prevented from going to the other's home/place of residence plus handovers can be done at school or through third parties if it can't be done at school or done in a public place where the children walk from one parent/third party to another.
Between the parents there may be verbal and written communication allowed, only direct written communication allowed, or only written communication through a third party e.g. family member, family friend.
Obviously this depends on the age and maturity of the children involved particularly if they are secondary age, and also the judge.