。
My daughter asked her mother to come with her.
Literal
Daughter [topic-は] mother [dative-に] accompany [てほしい] [quotative-と] asked.
同行する ('to accompany') is a formal Sino-Japanese verb. ~てほしい marks 'wanting someone else to do X.' The quotative と + 頼む reports the request. Three-layer embedding: the daughter is the speaker of the want, the mother is the target of that want, and 頼む is the outer reporting verb.