"He lied to my face for over a year after I thought it had ended."
So your H pretended to have ended a long-running affair, and he faked reconciliation, all the while seeing how devastated you were.
And now he's again saying he's ended the affair. And you're trying to believe him.
OP, I've been reading on infidelity boards for decades because my dad was a serial cheat who did untold damage to everyone, including us kids*. I wanted to understand what could lead to such destructive, cruel behaviour. I can tell you, there is ZERO chance that your H has ended the affair. ZERO. He's told you he loves her; he still sees her; and he has no intention of getting another job. All of this tells me that he's taken the affair underground. He's playing the penitent man to fool you. He's still lying his damned head off.
OP, this man is a cheater. That means he has the mindset of a cheater. Pretty much all cheaters have the same core profile: a craving for validation; a sense of entitlement; the ability to twist the truth in their own head so that they give themself permission to cheat; and the ability to lie to people straight in their face and to keep lying.
These things are very very deeply baked into the foundation of who these people are. They are weak, (self)deceptive, self-indulgent people who lie to get what they feel they 'deserve to have'. For them to change, they have to do ENORMOUS work on their very psychological foundation. A few counsellor sessions won't hack it.
You - and your children - are actually in quite some danger: there is a very substantial chance that he'll leave you, perhaps when the kids are old enough and the financial whack will be less for him.
You need therapy to process this and to help you think straight. If you won't do it for yourself, do it for your kids. They deserve to have a mother who is not riven with anxiety constantly, who becomes preoccupied and tense every time her H is late. They deserve a mother who is full of energy and joy and life and has ample emotional and mental time for them because she's happy.
I second the PP who suggested SurvivingInfidelity - they are pro-reconciliation but also know very well how narrow, painful, and excruciatingly difficult the path to true reconciliation is:
https://www.survivinginfidelity.com/forums/2/just-found-out/
I also recommend ChumpLady, who is not in favour of reconciliation because of how difficult it is and how rarely cheaters truly reform:
https://chumplady.com/arguments-against-reconciliation/
*Btw my mother initially thought we didn't know: we DID. Her lying to us, her self-deception, her endless hope that this time he really would change, and her excruciating fear - this was a major part of the trauma that was inflicted on us kids, who were quietly listening and watching and absorbing.