A.J.
I'm a huge disciplinarian, and the first to say to firm up. But I'm so glad you included your work hours here, so that I did not say that.
My daughter is almost 5. She's extremely sensitive by nature, and very well behaved, but I notice when I am distracted for long periods of time for deadlines (I work from home) or occasionally I travel for a few days, she gets much more sensitive, and her behavior slips. Likewise she reacts to the fact her dad is gone all the time, so I make allowances for that. Even though our budget it ridiculously tight, I am able to be a SAHM mom for now, for which I'm grateful, because I know it would be really hard on her to have me away more. Her more so than my other two. I can't imagine how she would be if I had your TOUGH schedule. That has to be so hard.
Obviously you are doing what you have to, and I hope you can get through this phase, because 10 hours at work plus school is not enough time with you for your daughter. She seems like she is hurting and acting out.
Think of a way to do something really special with her soon, and include in your time together talking about how proud you are of her when she acts well at school, and does not hit other kids. I don't believe in rewards only, or bribing kids to be good AT ALL, but I think in your case you should tell her that after x days (make it just one or two days) of doing great at school, you and she will (do something really fun together). Also be sure you have time EVERY day for just you and her to cuddle and check in where she doesn't feel rushed, and feels you have all the time in the world for her. Even 40 minutes of book reading can do wonders.
Lately, I've been sitting up (way too late) in bed with my daughter to talk about our day and letting her sleep there (hubs isn't home) because I haven't had time to focus on her much during the day. She really needs it, and even deprives herself of sleep waiting for that time together. All too soon she'll be independent and won't need me anymore, so I do whatever it takes-even leave my own shower or organizing or tasks to slip at times.
Firm discipline for hitting only works when you act firmly and immediately in that moment, not when you hear about it after school. Firm discipline for smarting off etc also only works within a loving close environment where the child is totally secure. She wouldn't be acting out at school if she was happy with firm boundaries at home. I wouldn't discipline her more firmly right now when she has so little time with you. Not that she should go without discipline (and kudos to you for not slacking), but her good times with you need to increase first. Get your relationship more stable, give her some new extra affection and privileges (with you) and then firm up if there is still trouble. Ideally, you want your husband and yourself both disciplining her for bad attitude right away when it happens, but be sure that's not all that's happening in her world. My son also gets a bad attitude when I slack or don't focus on him for a while, but I firm up when that happens because I'm home and can always handle it right away.
If my husband is home, I be sure to take my daughter with me on "special girl stuff" (even if it's just going to the grocery store) to make her feel special even if I'd rather have time alone, because it's the only time she gets my focus without the siblings. Have the hubs help with something to leave a little time for you and your daughter to go out to lunch or library or something in your off time. I'm sure you do, but do it even more. Good luck it's so hard!