Hi,
I am highly upset that someone with the children's agency would let your child get that upset. Someone there is not doing their job to "protect the children"! You need to talk to someone there to make sure that never happens again, because you cannot stop court ordered visits, so you have to make some changes to make visits easier for your child.
Now, unfortunately, it is up to you to try to fix this on your own.
1. Can you supervise the visits yourself, or a family member you trust? (I know, this will be hard, because she is supposed to be getting used to him, without you)
2. Is there a window to witness the visit and your child's reaction to her father? (Since apparently they dont know when to end the visit, you may need to)
3. For the comfort of you and your child, you need to explain to her that she NEEDS to visit with her father, without you and you will be waiting for her when she is done with her visit, which wont be long and its ok to be scared. You may have to explain this to her, and to yourself, every week. And dont forget to try to get her to talk about her visits right after the visit. (get her thoughts, feelings, and what she wants to do with her dad, like go out to eat, go to the zoo, or park/playground?) Keep conversation open, talk about everything.
4. Give her a picture of you that she can keep in her pocket to look at anytime she wants. Do this so she can look at it even when you are not in the room, so she can feel safe.
5. Find a dollie or teddy bear or even a small blanket (or a pillow case with your scent on it? whatever it takes) that makes her feel safe when you are not right by her side. Tell her that it is mommy's helper, to watch over her, keep her safe and be with her when you have things to do, even when your in another room. Then let her take it with her when she sees her dad, they cannot take it away from her!
I went throught the same thing with my son, now 8. He was 2 1/2 when his dad wanted to take him home with him, instead of visiting at my house. We went through court, he had 1 month of supervised visits, every saturday, at my parents house. (He drove 4 hours one way to do this). It broke my heart every second, knowing he was going to be taking my little boy so far away from me.
The second month, he had local overnight visits with my son. He drove up early every saturday, took him to a local motel, had to feed him, bath him and put him to bed at a decent time and learn to spend time with him alone. I demanded the local overnights because if there was a bad reaction, I wanted to be able to get to him quickly.
After those visits, he was able to take him home on Fridays and bring him home sundays. every other weekend. I was a real basket case for about 3 months until I could get used to him being gone, I had a real fear that he would not bring him home. All went well and he still shows up every other friday for our son.
It will turn out well and she does need her dad, good luck.