@Charnate,
Curious - how old is your daughter?
Quote: His relationship with his daughter has become very bad, they are always shouting at eachother and our daughter speaks to her dad very badly.
I am assuming she is under 18 - then she is a child and he is an adult. Although she should not speak to him badly - your husband is an adult and should not be shouting - I am guessing out of control at her.
Sorry I am focused on this because I think the most important thing here is the child in this situation.
About the kiss - no I would not tell him. You made a mistake and he does not seem like someone would be understanding about this - I do not think confusing this mistake to him would help your situation.
What you need to do - is sit down and think. Do you want to be with your husband - if yes, then you both obviously need some assistance. Talk with him and let him know you understand you both have been having issues - would he be open to couples counseling?
I think you two either need to work on it together - or this is going to continue going down hill or decide you are both better off a part.
The kissing of this man is just one symptom of the issues you are having.
And do not hang with this man any more - get your head on straight first.
-- One other thing - many of this was posted after you provided more details --- get yourself some help if you want to leave and you do not feel strong enough. You could seek a therapist just for your own benefit (doesn't need to be a couples thing) .
Here is something you could read that gives advice on thinking through staying or leaving
https://www.psychologytoday.com/us/blog/contemplating-divorce/201101/how-do-you-know-if-you-should-stay-or-go
And this helps you out if you want to leave - there is even a hotline if you feel you need immediate help.
https://www.verywellmind.com/how-to-leave-a-toxic-marriage-4091900