"I think I tend to think like AlbertBridge, in the sense that it is just sex".
It is not just sex though, it is indicative of a persons frustrations of being trapped in a situation where they feel they cannot escape. I am sure you feel the same. They are having sex with others and no longer willing to put their life on hold.
They have decided they can either remain, unhappily , where they are, until they can afford to leave or they can indulge and cheat , knowing that life can be short and they have no current financial means to leave but , it will hurt others , as yr DP/DH has clearly done. He has chosen the i have no money to divorce, life is too short option.
Yes, this behaviour hurts people, yes this behaviour is not acceptable in society, yes this behaviour is shitty but is ultimately a result of finances. Yes , this behaviour results from a wedding certificate that legally binds and precludes in many cases, financial independence when it fails. Marriages break down, sexually exclusive , long term relationships fail. People grow, people change, people grow apart.
When one person wants to move on but , are married and know that a whole load of other people ( wife and kids) are financially and now legally, solely dependant on them , they feel trapped. Being married benefits some but leaves others financially screwed. They now want their freedoms but equally they want to maintain a roof over their children's heads. There is a kind of responsibility in their recklessness. They cheat ! They have no option to easily release themselves from the financial burden but , they want the sex with someone else.
This is the time to move on OP. Hanging on to a relationship because it is either all you know, or you cannot afford to leave is not a reason to stay. Why do men beg for forgiveness? Because they cannot afford to house both you and yr shared children and , then himself comfortably. He has no financial option but to beg, you see it this as an act of contrition on his part and a new fresh start. He sees it , as self preservation.
Yr relationship has broken down and at this stage you need to look at being more financially self sufficient and independent. With maintenance from yr children's father and yr own income, you can house yourself, yr children and move on with yr life . You may still love him deeply but he has cheated. He is not on the same page as you anymore.
With kindness and upmost respect to you Op, you are hanging on to nothing. You are delaying the inevitable.