Hello, JCN, and welcome to Talking Point. I hope you find some good information, advice, and support here. Feel free to read around in current (and past) threads and join in, or not, as you wish.
I am sorry you're having a difficult time with your mum and that you have had to move her recently. That must be very stressful and a big transition for everyone.
How often are you visiting, and for how long? How far away from you is the new care home?
If you are visiting every day, or for long periods of time, you might consider cutting back on your visits until she settles better. I know it's counter-intuitive, but sometimes it works, especially if one family member is a "trigger" for upset feelings (not to say they have done anything wrong). Also if it's a distance for you to travel and then it's not a pleasant visit, that can make it much harder for you. On days when you don't visit, you might call or e-mail to check in with the staff to get information about how your mum is doing.
It also often helps to blame somebody else, as others have said. I tried deflecting my mother's angry questions about "why am I here? I want to go home! there is no reason for me to be here, I am fine!" (and for the record, she was not safe at home and needed the care home) by saying things like, the doctor said so and it's just until they get your medicines sorted out. Sometimes that worked and sometimes (in her clearer moments) that did not.
Eventually we settled on saying something like: I know you don't want to be here and are upset. I'm sorry you are so upset. Nobody wanted it to be like this, but this is how it has to be for now.
As she got settled, she got calmer and she has mostly stopped talking about "home."
Also let us reassure you that if your mum has been placed in a care home for her own safety, as it sounds like, then you have done the right thing and part of you coping is holding tight to the knowledge that she can't go back to the way it was.
You might also learn some strategies to deflect or distract her comments. One is physically leaving the room (have to go to the toilet, mum!). Sometimes this is enough to change the subject and it gives you a nice, private room with a locked door where you can take a deep breath. (I went to the toilet A LOT at the beginning, but my mother has short-term memory issues so she never noticed!) If it's really bad, I stay there for a few minutes and text a friend or relative.
Another strategy is to not visit alone. My mother was very unpleasant to me if I was there alone but fine if my husband came along. She's better now but I still don't take her out by myself.
And I also recommend having an exit strategy. Whenever you've had enough, or it's just time to leave, then leave and make it quick and clean. If good-byes are very bad, leave without saying good-bye. You might also time your visit for shortly before a meal or activity so they have something to distract them right away.
Wishing you all the best with this difficult situation.