Social worker of many long years here. 90 percent of those years spent in child protection. Studied as a single parent with no help from family or her dad. So I feel able to say this is completely normal op. Not right or ideal but normal non the less. I have worked with many many social workers who are parents and I don't know one who would not admit to struggling in a similar way. The job is utterly draining physically and emotionally and when you get home no matter who you go home to your brain and body is done.
Now so real advice if you want it-
Avoid CP work once qualified if at all possible. If you are adamant this is the area you want to work in then you need to set out your boundaries from the get go and you need to be strong in always prioritising yourself and your child. You need to find the confidence to say no and to make the job work for you. You can and you must be ready to walk away if the day is getting too much and to be comfortable in telling your manager that you are picking your dd up and you will work flexibly. This job can be done in various ways and at odd times so don't let them talk you into working office hours. I work around my dc. I work through the night if I want to. I do all my visits on one or two days then I work from home and do the hours round my dc school and activities and time with them.
Once you are confident and have gained some experience go agency. Don't worry about completing your asye if that time comes before you have finished it. It's not mandatory and you won't struggle to get a job without it. The agency pay and flexibility is worth it.
For the here and now: speak to your manager and tutor and see if their is scope for more flexibility or condensing your hours or time left. It's there job to support you and make it possible for you to succeed. There is a national shortage of people training so they will want to hold onto you.
As bad as it sounds you do have the upper hand now and upon qualification because of the staff shortage so you can ask for what you need and want with confidence.
Check you are getting everything you can financially and get help with this from uni and from a knowledgeable person at your placement. There is help out there but you have to seek it out. This may mean you can give up one of your jobs at least. If you speak to your manager and tutor about what you are going through due to financial hardship I'd be surprised if they didn't have some suggestions and solutions. It will not be the first time they have come across this.
Accept you don't have to have all the answers to problems on your cases or in life. You will never know what you have to do 100 precent of the time. I ask daily what I'm supposed to do with this or that as there is so much to learn. Even years in I have to ring people about how to fill out some paperwork I can't remember ever doing. This will help you as you can let go and don't spend time thinking about or studying this at home. You only need to pass the course even if it's bare minimum marks. You don't really start learning till you're actually properly doing the job anyway. No one cares if you got a pass or a first in the real world and you will get a job regardless. Give yourself a break.
Get some St John's wart extra strong from boots and take daily.
Take a multi vitamin daily and force yourself to eat the fruit you take to work before you go to mcds with everyone. Even if you still go to mcds you will be fuller so get less, saving money and feeling better as a consequence. Seems bizarre advice but we know all social workers eat terribly.
Find stuff to do with dd that you enjoy. I don't enjoy watching the endless made up dances or pretending to be Ken and Barbie but I do enjoy singing to my favourite songs and asking my dc to judge whether I'd get the X factor golden buzzer and I do enjoy doing face paints in the dc or puzzle books. Just examples because ding these things you enjoy will benefit your dd and you can be genuinely present rather than half arsed. As you know the less you fill her cup the more likely she will demand your attention and do so in annoying ways. It's quality not quantity. You can give your dd time in your terms and chances are she will love it. You can do her choice of thing too occasionally but your not damned to only spending time with her that involves what she likes and most adults find mind numbing. Some people think they have to play Barbie's or do dc led activities in order to fill dc cup but you can do this while doing something you like and involving her.
Some nights bring a treat home for dd. Not something expensive, we are not buying out dc here, I grab a small piece of cake while I'm getting my Costa for example and take it home so dc know I have been thinking about them. We then eat this together and watch something funny. Keep it simple and chill coz you're burnt out.
Stay off your phone till her bedtime if possible some nights. The reason I say this is because from experience we go down a rabbit hole and struggle to get off it. Plus it makes you more tired. Our brains crave that doom scrolling time after work but it's lethal and prevents you from getting stuff done or interacting with dc.
Put music on while you cook dinner and have dd sit and chat and sing along with you while you cook. Get her to help you. Laugh at each other's mistakes or funny shaped veg. Basically fake it till you make it. It wil feel unnatural and another chore at first but it gets easier over time.
Find out if any of DDs friends attend clubs on the days you can take her. Drop her off. Go for a coffee or chat to another mum while you wait to pick her up. Ask her after how it went, what she learnt, what her friends did and said and so on. This not only gives you a small break when she's busy enjoying herself but also gives you something to talk about that makes her feel like you're interested. No kid cares about their parents asking about school really, so you're both just going through the motions, but they do care if it's an interesting activity or involves their friends.
Accept it's hard right now but with a little extra effort with your dd you will feel less guilty. The guilt is adding to your feelings of burnout. Guilt is a draining emotion. It doesn't have to be all or nothing as I suggested above make the time with dd easy for you both and not always one sided. The more you give this quality time the less time she will spend dancing in front of you to get snippets of attention.
Oh and in the day you are not working and dd is at her dads. Don't do the other stuff whatever that is. Just go to bed or something fr yourself and rest rest rest. The house can be a bit of a shit hole for a bit just do the basics and it's fine.