Talk calmly in private with your girlfriend. I think that the 4 year old fears that since her father was replaced that she might be next? You cannot be her father, but instead her friend . The only time that a child will accept you as a father figure is when they are 3 years of age of younger( according to Doctor Phil’s observation over the years.). Also they will NOT accept discipline well from an outsider as they see you as.( therefore only the mother can discipline her). Best way to handle it is to talk openly to your girlfriend. ( perhaps another female Grandmother , Aunt can take up the time with her as a outlet. But not sure how she would handle that.