Hello. I believe I can be of help to you with this issue.
It would appear that the only hope for these kids is long-term placement in a stable firm-but-kindly foster care setting or group home as a first step to rehabbing the situation, the mother, etc. Kids behave according to their immediate environment as do adults; take them out of bad situation and put them into a better situation; but then, return them, and they will always revert to the behavior that is reinforced or supported by whatever environment they are in. So if the kids revert back to old behavior when they go back home after a group home stay, it is clear that the mother is not skilled or competent enough to support and reinforce appropriate behavior---she is doing whatever she did
before to support misbehavior.
The problem is that the 'system' is loath to take kids away from dysfunctional parents long term. These parents are obviously, unskilled, beset with so many problems of their own, that they are unwilling or unable to parent. So this would be my recommendation: Take the kids out of the home and insist that for mom to get them back, she get into a good alcohol counseling program, go to vocational rehabilitation services and find out what can be done to get her job skills improved; get the courts to sit on the dad's paycheck and encumber it so he pays his child support.
Frankly, no one is really doing anything to help ardently, consistently. Everything appears to be done in a piecemeal, reactive fashion. For example, I can tell you that the mental health services this woman has available to her (individual counseling several times) was probably not competent counseling because she obviously wasn't helped. A good therapist knows motivational interviewing skills sufficient to draw her in and then, work with her effectively. Mom would get the kids back on trial periods of a day or two at a time and be expected to SHOW she has acquired better parenting skills e.g., setting limits, chore charts, homework, reinforcing compliance with access to video games and TV or other preferred activities. Has anyone assessed whether mom has the skills to even be trying to do home schooling? Can she give a lesson and does she have enough skill of her own to tutor her kids? Basic questions like these need to be asked and assessed. She would be strenuously discouraged from 'dating' and diverting her attention away from her kids until the home showed behavioral stability---both on her part and the kids'. So here is a solid, practical question: If one looked at the therapy "notes" from all of her counseling visits, I would lay heavy bets that the counselor never once considered tacking mom's problems is a global, systemic way. The notes would show that session after session, mom's personal problems, her 'slice' of the big picture view of the situation was all that was ever discussed. No broad, family plan was in place, no parenting skills training, no plan to extract full child support was put into action, no family therapy sessions occurred with the kids, no behavioral contracting on behalf o the kids occurred, etc. Mom was never assessed in terms of her skills to home school, etc., etc.
I'm not sure about the best way to rework this situation e.g., perhaps find a school psychologist or counseling in the kids' school to meet with you and set up a strategy to get something broader, more systemic put together to help the family. One angle: See if there are any legal charges against the mom or dad and work the case through a family court program, if one exists in your locale. A more serious and concerted effort may not take place unless there is a significant legal event e.g., one of the kids burgles a home and gets arrested or the girl gets picked up for accepting money for sexual favors, shoplifting, etc. This would be the time to get the school counselor/psychologist, anyone else concerned etc., to sit down with the prosecuting attorney and lay out what the court might be able to demand and enforce.
As is, little can be done in all likelihood to change things; the situation actually 'needs' a more serious crisis to occur. What do you think?