My OH works 6-4 days M,T, & W, comes home, plays with Cay a little bit, and is always there while I bath him/massage him (plays with him in the bath and keeps him occupied while I get his PJs on), then we put him to bed. Then he normally deals with dinner, although I do it on the odd occasion (spot the cook in our family... Lol) and I wash all Cays bottles/bowls/etc and get them sorted for the next day.
Thursday and Sunday, he's off work, so we both do stuff with Cay, and he'll have him on his own for an hour or so while I pick some housework to get done.
Fri and Sat, he works 5pm-4am, so doesn't get up until about midday. He'll take Cay for a coupla hours while I have some me time (like a nice long bath instead of a two-minute shower), and feed Cay either his lunch or tea depending.
So basically when OH is home, Cays time with us is pretty even, although I have more time with him at the minute because I'm still on maternity.
Once I go back to work in a coupla weeks, Cay will be in childcare two days a week, with me one day, OH one day, both of us one day, and split between for the other two days. Xx
Okay, so I wrote all that ^ this morning, and didn't get round to posting. Now I've seen your reply about how your life works.
So, my advice... Sit down with OH and seriously talk to him. Some time when your little girl isn't around. Arrange for her to be in nursery a bit longer, or go to granny's or something so you can really talk and clear the air.
Be calm, and tell OH what you've told us here. Lay out his shift patterns, and sleeping/"me" times, your work and sleep times, and explain how your daughter is up all night and you get no real sleep. Explain to OH that you're exhausted, and beginning to resent both him and your girl, and that you've reached the point where something drastically need to change.
You say your daughter was unplanned and unwanted. Was she always unwanted? Why did you keep her? Or do you say she's unwanted now because you're just physically and emotionally exhausted?
Cay was a surprise, but nothing in the world could make me want to change anything now he's here. Have you actually bonded with her, or is she just an inconvenience?
If she's genuinely unwanted, you may wish to think about adoption, so she can have the loving upbringing she deserves. If you are physically and emotionally unable to provide this, then she is missing out. There's no shame in admitting that you're struggling, and it may be a route you wish to consider.
If you actually do want her, but are just clouded at the moment due to your seemingly selfish partner, then you need to think about wether he has a place in your lives from here on. Yes, it would be hard to bring her up on your own, but at least you would KNOW you're on your own, and not continuously living in hope of help from someone who is apparently not prepared to give it.
You need to explain all this to your partner, and see where exactly he stands.
If you feel this is a conversation you can not have with him, then, I guess you have your answer there about where your future leads as a couple.
Ultimately, it seems to me that you are just completely drained, and in need of some help quick.
I would strongly suggest speaking to a professional, both alone and as a couple (if you decide to stay as a couple). They would be able to help you with time management, and communication, if nothing else.
I sincerely hope you find the answers you need xxx