Hello and thanks for your question.
In theory, your partner as the child's father could apply for a child arrangements order under section 8 of the Children Act 1989, asking the court to order that his daughter live with him - but the court is very reluctant indeed to change a child's residence unless there are very strong reasons indeed. The wishes of an 8 year old child would not be a strong enough reason. The court decides all issues concerning a child on what is best for the welfare of that child. If your partner truly felt that the welfare of his child was so bad being cared for by her mother and stepfather, then he should apply to court. The court would order a welfare report to look into the situation in detail. The court case would take several months. The court form needed is a C100, here:
The court now requires the parties to have attempted mediation before it will consider an application to court. Here's where to find a local family mediation service:
I think your partner would also benefit from some face-to-face legal advice. Here's where to find a specialist family law solicitor:-
Instead of, or in addtion to, making an application to court, your partner could consider contacting social servvices, if he believes that his daughter is being mistreated in her mother's care.
I hope this helps and I wish you the best of luck.
Thanks and best wishes...