Do you have a previous thread? This might help with context and advice.
Do you have a family support worker? These are in shorter supply now but it might be something you can access.
How old is your child(ren)? Are they in any childcare at all? Are you working?
I was also referred (quite a lot of years ago now) due to concerns mainly surrounding messy house. I was struggling with what I didn't know at the time was undiagnosed ADHD, I recognised depression and overwhelm - but what happened was that basically the health visitor would come over and she was concerned about the state of the house, and the fact that DS1 was always hungry when we went to the Children's Centre stay and play. She had shared these concerns with me. I tried as best I could to address them, and she came over again and said that it just looked the same and she would make the child protection referral. I was so so upset because I had tried and I thought there was a difference, but apparently it hadn't made any difference to her.
I share this because you mentioned putting effort into certain areas and ignoring the rest. That is a brilliant start! And a really good idea to prioritise, especially when the whole is overwhelming. But if you are a chronically messy person (me...) then you might have a completely different idea to your health visitor, or a social worker, about what is normal and maybe also about what is important.
What happened next for me was a CAF meeting (I think they now call this Team Around the Child - it might have another name.) and the social worker was there. He wasn't scary. I was really scared, and this is a kind of almost primal fear. I've never felt anything like it. I was doing my best, but somehow it wasn't good enough and they had the power to make decisions without me about the most precious and important thing. It's like white noise in your head all the time, which doesn't help. But they do not want to take your child into care. Frankly, it's expensive and complicated, they also understand that it is distressing for the child. Be open to working with them and whatever they have to say, and they will probably appreciate this. Social workers get a lot of defensiveness, anger, refusal to cooperate from parents, because nobody is their best self when they are scared, and because some families who are struggling have their own trauma related to social workers possibly from their own childhood. So they are generally relieved and happy if you are willing to talk to and work with them. Try to keep this in mind and not let the fear dominate.
In the meeting, I explained the two instances with my health visitor, and expressed my own confusion and dismay about having tried but not actually getting it right, and the social worker asked if I would like a copy of the guidelines that they use to assess the cleanliness and state of a house. I said yes and I received two copies. The guidelines were eye opening to me because some things I hadn't even realised were not normal. One example that I remember is leaving washing up to be done when I next needed the item, instead of doing it regularly. (Every day or after meals). This is how my own mother had coped when I was growing up, so I thought that is how you did washing up. Maybe that's not a specific misunderstanding that you have, but it is an example of one of the things that my health visitor had immediately spotted, whereas I had focused on making sure that the piled up washing didn't have old food in it, for example. And another was noticing that the toilet was stained beneath the water line. That's how my toilet looked, even when I cleaned it, so I thought I couldn't do anything about it, but I have learned that if that layer is not allowed to build up, then it is possible to keep it away, and it was simply that I should have been cleaning it more regularly so as not to let it build up. (There are also methods to remove stained limescale that is already there).
Anyway, having the checklist was useful because it was objective and it gave me specific things to work on. I actually never saw the social worker again and they closed the case. My health visitor went on maternity leave and I got a different one who I found easier to talk to (pure chance). They felt that because I was engaging with the support services, and my child was healthy, they did not need to follow up.
In terms of getting on top of the housework, I would recommend two resources that actually work when you're in utter crisis and drowning in it. My favourite is Dana K White's A Slob Comes Clean, or her book How To Manage Your Home Without Losing Your Mind. The other one which is relevant for somebody who is in crisis is How To Keep House While Drowning by KC Davis. Both of these authors also have instagram, blogs, podcasts, tiktok, whatever appeals to you. I've found putting the podcasts on while I do cleaning or tidying helps me.
There are other systems that people often recommend such as the organised mum method or FLYlady, but IME, those things are useful only if you keep doing them, and keeping getting back up and doing it when it felt impossible was my road block. Both Slob and Drowning address this part of the issue specifically.