sorry, but how did you "notice" the card, handbag and champagne in his kitchen, before you rang the bell to confronthim over htis?
you would absolutely be unreasonable to withhold ocntact over this.
he has doen nothing to make you suspect he woudl not make contact all about the children.
he slept with someone else, in his house, when the children were with you.
he is an adult. he is allowed ot do this. it is no business of yours. you have no reason to suspect this woman oul even be there at contact time, and even so, it is not up to you what your h does with the children at contact, or who he introduces them to.
you might have some rigth to ask that he waits ot introduce them, but they are his children too, and he has as much say as you in this.
you would, quite rightly, be incensed if he suggested what oyu did with the children while they are with you, and where you took them, who they spoke to etc.
your children have every right to see their father. on what you have posted here, you have no basis to withhold contact. you being upset at somehtign he has done is not a reason to make sure he cannot see the children.