If a parent has lost custody of one child from another relationship is it harder for them to get custody of another?

It depends on why she lost custody. If it was because of an abuse, neglect or substance abuse problem or because she has mental health problems that make her unfit as a custodial parent then it probably would make it more difficult to get custody of a child from a current relationship. The difficulty is not because she has had custodial problems with another child but because she has very serious underlying issues that make her unfit or nearly unfit to have unsupervised contact with any child. When an issue like that comes up, most attorneys will look at the court pleadings and orders involving the other child to determine what mom's serious problem is. You will need to prove that whatever the problem is, that problem has a direct and adverse effect on your minor child.

Most likely, yes. The basis for such limited visitation with her other child will be a key issue.