。
She seems to be struggling with a handicap from her family circumstances.
Literal
She [topic-は] family-related handicap [in-に] is-suffering [seems-ようだ].
家庭的 typically means 'domestic, homely, family-oriented,' but here it's modifying ハンディキャップ ('handicap') with the sense of 'related to the family / arising from one's home environment' — a slightly less common usage that points at family-circumstance disadvantages: difficult home life, financial hardship at home, lack of family support. ~に苦しむ ('suffer from') takes the source of suffering with に. ~ようだ ('seems, appears') is the speaker's inference based on observation. ハンディキャップ is an English loanword, less euphemistic in Japanese than in English where it's now mostly avoided.