At home, because of the reddish hair and freckles, Mother contemptuously named the child 'Carrot' and makes everyone call them that too.
Literal
Home [at-では] reddish hair [and-と] freckles [because-のせいで], mother [subject-が] contempt [object-を] putting-in 'carrot' [quotative-と] named, everyone [to-にも] so call-[causative-させています].
A passage from or about Jules Renard's novel 'Poil de Carotte' (にんじん). 赤味がかった ('reddish, tinged with red') uses the ~がかった suffix meaning '-ish, having a tinge of.' 侮蔑を込めて ('loading contempt in') is vivid — 込める means to put something into a container, here loading words with contempt. 呼ばせている is causative + continuous: the mother actively makes everyone continue using the nickname.