When my father got sick, we had to get by on a small income.
Literal
Father [subject-が] sick became [because-ので], we [topic-は] small income [with-で] living go-must was.
暮らしていかなくてはならなかった is a dense construction: 暮らす (live) + ていく (continue) + なくてはならない (must) + かった (past). It means 'had to continue living.' 少ない収入で marks the meager means.